THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Excerpts
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of $\zeta$. The right-hand sides have, on the other hand, been defined for all values of $\zeta$, real or complex. We are therefore naturally led to adopt the formulae (1) as the *definitions* of $\cos \zeta$ and $\sin \zeta$ for all values of $\zeta$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations (2)--(6); and so all such relations hold also for the generalised trigonometrical functions defined in this section.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
where $-1 \leq x \leq 1$: each of these formulae also is ‘completely’ true.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The function is discontinuous for $\theta = (2k + 1)\pi$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
It is evident that $\cos \zeta$ and $\sec \zeta$ are even functions of $\zeta$, and $\sin \zeta$, $\tan \zeta$, $\cot \zeta$, and $\cosec \zeta$ odd functions.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Let $z$ be any complex number, and $h$ a real number small enough to ensure that $|hz| < 1$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
_h0 (1 + hz)h = z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
ddt(1 + tz)^m = mz(1 + tz)^m-1
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Here both $(1 + tz)^{m}$ and $(1 + tz)^{m-1}$ have their principal values
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
for all values of $m$, real or complex, and all values of $z$ such that
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
We shall call this particular value of $\Log \zeta$ the **value**. When $\zeta$ is real and positive, $\zeta = \rho$ and $\phi = 0$, so that the principal value of $\Log \zeta$ is the ordinary logarithm $\log \zeta$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The equation $\Log z^{m} = m\Log z$, where $m$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The definition, although perfectly legitimate, is futile because it does not really define a new idea at all.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
This is a further generalisation of De Moivre’s Theorem
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if $z$ is given, so are $x$ and $y$, and conversely: to assign a value of $z$ is precisely the same thing as to assign a pair of values of $x$ and $y$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Since $|\zeta | = \rho$, and the different angles $2k\pi + \phi$ are the different values of $\am \zeta$, we conclude that every value of $\log |\zeta| + i\am \zeta$ is a value of $\Log \zeta$; and it is clear from the preceding discussion that every value of $\Log \zeta$ must be of this form.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
An equation such as (1), in which every value of either side is a value of the other, we shall call a *complete* equation, or an equation which is *completely true*.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
It would not be unnatural to suppose that, conversely, to any given value of $\zeta$ correspond infinitely many values of $z$, or in other words that $\exp \zeta$ is an infinitely many-valued function of $\zeta$. This is however not the case, as is proved by the following theorem.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
It might seem natural, as $\exp \zeta = e^{\zeta}$ when $\zeta$ is real, to adopt the same notation when $\zeta$ is complex and to drop the notation $\exp \zeta$ altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol $e^{\zeta}$: we shall find then that $e^{\zeta}$ represents a function with infinitely many values of which $\exp \zeta$ is only one.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
We conclude that *$a^{\zeta}$ is infinitely many-valued unless $\zeta$ is real and rational*. On the other hand we have already seen that, when $\zeta$ is real and rational, $a^{\zeta}$ has but a finite number of values.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Explain the fallacy in the following argument: since $e^{2m\pi i} = e^{2n\pi i} = 1$, where $m$ and $n$ are any integers, therefore, raising each side to the power $i$ we obtain $e^{-2m\pi} = e^{-2n\pi}$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in % [examples:lxxxvii]Ex. lxxxvii%. 21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have expressed the circular functions of $\zeta$ in terms of $\exp i\zeta$, and that the logarithm is the inverse of the exponential function.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
Moreover we saw in [§]191 that the series on the right-hand side remains convergent (indeed absolutely convergent) when $z$ is complex. It is naturally suggested that the equation (1) also remains true, and we shall now prove that this is the case.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
We shall in fact prove rather more than this, viz. that (1) is true for all values of $z$ such that $|z| \leq 1$, with the exception of the value $-1$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The sums of the series, for other values of $\theta$, are easily found from the consideration that they are periodic functions of $\theta$ with the period $2\pi$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
This is a generalisation of the result proved in [§]208 for real values of $z$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
A more complete discussion of the binomial series, taking account of the more difficult case in which $|z| = 1$, will be found on pp. 225 *et seq.* of Bromwich’s *Infinite Series*.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
If $f(z)$ is a function of the complex variable $z$, we call the curves for which $|f(z)|$ is constant the *level curves* of $f(z)$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
This projection gives a map of the sphere on the tangent plane, generally known as the *Stereographic Projection*.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
To one value of $Z$ corresponds one of $z$, but to one of $z$ infinitely many of $Z$.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.
Equations
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
z = x + iyThe complex variable z is written as x plus i times y, with x and y real.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
|z| = \sqrtp{x^{2} + y^{2}}The modulus of z is the square root of x squared plus y squared.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\am z = \arctan(y/x)The amplitude of z is the arctangent of y over x.
- This equation is in THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE (THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\ OF A REAL VARIABLE)
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\int_{C} \{g(x, y)\, dx + h(x, y)\, dy\}The real curvilinear integral of g dx + h dy along the path C is defined as the ordinary integral obtained by substituting the parametric equations of C.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\int_{C} f(z)\, dzThe integral of f(z) dz along C is defined as the real curvilinear integrals of (u dx - v dy) plus i times those of (v dx + u dy), where f = u + iv.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log \zeta = \int_{C} \frac{dz}{z}The general logarithm of zeta is the integral of dz/z along any curve C from 1 to zeta that does not pass through the origin.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\zeta = \rho(\cos\phi + i\sin\phi)A complex number zeta is written in polar form with modulus rho and amplitude phi.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log \zeta = \log \rho + i\phiWhen the path of integration is a straight line from 1 to zeta, the value of Log zeta is log rho plus i phi.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log \zeta = \log \rho + i\phiThe principal value of Log zeta, written log zeta, equals log rho plus i phi, with imaginary part between -pi and pi.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log |\zeta| + i\am \zeta = \log \rho + i(2k\pi + \phi)The general value of Log zeta is log of the modulus of zeta plus i times the general amplitude, where k is any integer fixed by the path.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log z_{1} z_{2} = \Log z_{1} + \Log z_{2}The general logarithm of a product equals the sum of the general logarithms; every value of each side is a value of the other, so the equation is completely true.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log z_{1}z_{2} = \log z_{1} + \log z_{2}The principal-value form of the logarithm of a product is not true in all circumstances; the book gives z1 = z2 = (-1 + i sqrt 3)/2 as a counterexample.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log z^{m} = m\Log zFor integer m, the general logarithm of z to the m equals m times the general logarithm of z; this is not completely true, since only one direction holds for all values.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log (1/z) = -\Log zThe general logarithm of the reciprocal of z is minus the general logarithm of z; this equation is completely true.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\Log e^{\zeta} = (1 + 2m\pi i)\zeta + 2n\pi iThe general logarithm of e to the zeta has values (1 + 2m pi i) zeta + 2n pi i, for integers m and n.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
z = \exp \zetaz is defined as the exponential of zeta when some value of Log z equals zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\exp \zeta = e^{\zeta}When zeta is real, the complex exponential exp zeta equals the real exponential e to the zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\exp (\xi + i\eta) = e^{\xi} (\cos\eta + i\sin\eta)The exponential of xi plus i eta has modulus e to the xi and amplitude eta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
f(\zeta_{1} + \zeta_{2}) = f(\zeta_{1}) f(\zeta_{2})The exponential function satisfies the functional relation f(zeta1 + zeta2) = f(zeta1) f(zeta2) for complex arguments.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
a^{\zeta} = e^{\zeta\log a}For positive a and real zeta, the general power a to the zeta equals e to the zeta log a, the definition used in the earlier chapter.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
a^{\zeta} = \exp (\zeta\Log a)The general power a to the zeta is defined as exp of zeta times any value of the logarithm of a, for any nonzero complex a and zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
|a^{\zeta}| = e^{\xi\log \sigma - \eta(\psi+2m\pi)}The modulus of a general power a to the zeta depends on the integer m unless eta is zero, so the general power has infinitely many values.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
e^{\zeta} = e^{\xi-2m\pi\eta} \{\cos(\eta + 2m\pi\xi) + i\sin(\eta + 2m\pi\xi)\}The general value of e to the zeta, with integer m, is given in terms of xi and eta; its principal value is exp zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\zeta = \Log_{e} z = \frac{\log |z| + (\am z + 2m\pi)i}{1 + 2n\pi i}On the second definition the logarithm to base e of z is doubly infinitely many-valued, given by this formula for integers m and n.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
a^{\zeta} × b^{\zeta} = (ab)^{\zeta}The product of a to the zeta and b to the zeta equals (ab) to the zeta; this is completely true but not always true of principal values.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
a^{\zeta} × a^{\zeta'} = a^{\zeta+\zeta'}The product of a to the zeta and a to the zeta-prime equals a to the zeta plus zeta-prime; this is not completely true but holds for principal values.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\pi f(x) = p\pi + (q - p)\Imag(\log x)The function f of the real variable x equals p for positive x and q for negative x, since Im(log x) is 0 or pi.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\exp (\xi + i\eta) = \exp \xi(\cos\eta + i\sin\eta)The exponential of a complex sum equals e^xi times (cos eta + i sin eta).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\exp (i\eta) = \cos\eta + i\sin\etaThe exponential of i times eta equals cos eta plus i sin eta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos\eta = \tfrac{1}{2} \{\exp (i\eta) + \exp (-i\eta)\}Cosine of a real angle written as the average of exp(i eta) and exp(-i eta).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin\eta = -\tfrac{1}{2}i\{\exp (i\eta) - \exp (-i\eta)\}Sine of a real angle written in terms of exp(i eta) and exp(-i eta).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos\zeta = \tfrac{1}{2} \{\exp (i\zeta) + \exp (-i\zeta)\}Defines cos zeta for every complex zeta by this exponential expression; it agrees with the elementary cosine for real zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin\zeta = -\tfrac{1}{2}i \{\exp (i\zeta) - \exp (-i\zeta)\}Defines sin zeta for every complex zeta by this exponential expression; it agrees with the elementary sine for real zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\tan \zeta = \frac{\sin \zeta}{\cos \zeta}Tangent defined as sine over cosine, for complex argument.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cot \zeta = \frac{\cos \zeta}{\sin \zeta}Cotangent defined as cosine over sine, for complex argument.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sec \zeta = \frac{1}{\cos \zeta}Secant defined as the reciprocal of cosine.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cosec \zeta = \frac{1}{\sin \zeta}Cosecant defined as the reciprocal of sine.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos \zeta = \tfrac{1}{2} \{t + (1/t)\}Cosine written in terms of t = exp(i zeta).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin \zeta = -\tfrac{1}{2}i \{t - (1/t)\}Sine written in terms of t = exp(i zeta).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos^{2} \zeta + \sin^{2} \zeta = \tfrac{1}{4}[\{t + (1/t)\}^{2} - \{t - (1/t)\}^{2}] = 1The sum of the squares of cosine and sine is 1 for all complex zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin (\zeta + \zeta') = \sin\zeta \cos\zeta' + \cos\zeta \sin\zeta'Sine of a sum of two complex arguments, in the same form as elementary trigonometry.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos(\zeta + \tfrac{1}{2}\pi) = -\sin\zetaShifting the argument by a right angle turns cosine into minus sine.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin(\zeta + \tfrac{1}{2}\pi) = \cos\zetaShifting the argument by a right angle turns sine into cosine.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cosh\zeta = \tfrac{1}{2} \{\exp \zeta + \exp (-\zeta)\}Defines the hyperbolic cosine for all complex zeta by this exponential expression.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sinh\zeta = \tfrac{1}{2} \{\exp \zeta - \exp (-\zeta)\}Defines the hyperbolic sine for all complex zeta by this exponential expression.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos i\zeta = \cosh \zetaCosine of i zeta equals the hyperbolic cosine of zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin i\zeta = i\sinh \zetaSine of i zeta equals i times the hyperbolic sine of zeta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cosh 2\zeta = \cosh^{2} \zeta + \sinh^{2} \zetaDouble-argument identity for the hyperbolic cosine, obtained by transforming the cosine double-angle formula.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\zeta = 2k\pi ± \arccos aSolutions of cos zeta = a for real a with -1 <= a <= 1 (real branch).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\int \frac{dx}{x^{2} + \alpha} = \frac{1}{\sqrt{\alpha}} \arctan \frac{x}{\sqrt{\alpha}}Integral of 1/(x^2 + alpha) for alpha > 0 is an inverse tangent.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\int \frac{dx}{x^{2} + \alpha} = \frac{1}{2\sqrtp{-\alpha}} \log \left|\frac{x - \sqrtp{-\alpha}}{x + \sqrtp{-\alpha}}\right|Integral of 1/(x^2 + alpha) for alpha < 0 is a logarithm.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\arctan \left(\frac{x}{\alpha}\right) = \frac{1}{2i} \log\left(\frac{x - i\alpha}{x + i\alpha}\right) + CThe formula the book proposes by analogy (i alpha written for alpha), with C a constant; the book then tests whether it holds.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\arctan x = \frac{1}{2i} \Log\left(\frac{1 + ix}{1 - ix}\right)Standard connection between the inverse tangent and the principal logarithm, true for real x; checked by putting x = tan y.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\exp z = 1 + z + \frac{z^{2}}{2!} + \dotsThe exponential function equals its power series for all complex z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
F(z) F(h) = F(z + h)The series sum F satisfies the functional equation of the exponential.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
f'(y) = \lim_{k \to 0} \frac{f(y + k) - f(y)}{k} = if(y)Differential equation satisfied by f(y) = F(iy): its derivative is i times f.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
f(y) = \cos Y + i \sin Yf(y) has modulus 1 and so can be written as cos Y + i sin Y for some angle function Y.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
F(iy) = \cos y + i\sin yThe exponential series at a pure imaginary argument gives cos y + i sin y for all real y.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
F(x + iy) = F(x) F(iy) = \exp x(\cos y + i\sin y) = \exp(x + iy)The exponential of a complex number x + iy is exp x times (cos y + i sin y).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos z = 1 - \frac{z^{2}}{2!} + \frac{z^{4}}{4!} - \dotsPower series of cosine, valid for all complex z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin z = z - \frac{z^{3}}{3!} + \frac{z^{5}}{5!} - \dotsPower series of sine, valid for all complex z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log(1 + z) = z - \tfrac{1}{2} z^{2} + \tfrac{1}{3} z^{3} - \dotsThe principal logarithm of 1 + z equals its logarithmic series for |z| <= 1 except z = -1.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log \left(\frac{1}{1 - z}\right) = -\log(1 - z) = z + \tfrac{1}{2} z^{2} + \tfrac{1}{3} z^{3} + \dotsLogarithmic series obtained by replacing z with -z in the previous series.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log(1 + z) = \int_{C} \frac{du}{u}The principal logarithm of 1 + z as an integral over the straight line C from 1 to 1 + z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\arctan z = z - \tfrac{1}{3}z^{3} + \tfrac{1}{5}z^{5} - \dotsPower series of the inverse tangent for |z| < 1, obtained from the logarithmic series.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\cos\theta - \tfrac{1}{2} \cos 2\theta + \tfrac{1}{3} \cos 3\theta - \dots = \tfrac{1}{2} \log(4\cos^{2} \tfrac{1}{2}\theta)Sum of the cosine series for log(1 + e^(i theta)) on the unit circle, valid for theta not an odd multiple of pi.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\sin\theta - \tfrac{1}{2} \sin 2\theta + \tfrac{1}{3} \sin 3\theta - \dots = \tfrac{1}{2} \thetaSum of the sine series for -pi < theta < pi; the sum is a periodic, discontinuous function of theta.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\log(1 + hz) = hz - \tfrac{1}{2}(hz)^{2} + \tfrac{1}{3}(hz)^{3} - \dotsFor |hz| < 1, the logarithm of 1 + hz is given by its power series in hz.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\frac{\log(1 + hz)}{h} = z + \phi(h, z)The quotient log(1 + hz)/h equals z plus a remainder term phi(h, z) that tends to zero as h tends to zero.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\lim_{h\to 0} \frac{\log(1 + hz)}{h} = zThe limit as h tends to zero of log(1 + hz) divided by h is z.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\lim_{n\to \infty} n\log \left(1 + \frac{z}{n}\right) = zTaking h = 1/n, n times the logarithm of (1 + z/n) tends to z as n tends to infinity.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\lim_{n\to \infty} \left(1 + \frac{z}{n}\right)^{n} = \lim_{n\to \infty} \exp\left\{n\log\left(1 + \frac{z}{n}\right)\right\} = \exp zThe limit of (1 + z/n) to the power n as n tends to infinity is the exponential exp z, a generalisation of the real case.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\frac{d}{dt} \{\log(1 + tz)\} = \frac{z}{1 + tz}The derivative with respect to t of log(1 + tz) is z divided by 1 + tz.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
(\psi' + i\chi') \exp(\psi + i\chi) = \phi' \exp\phiFor a complex function phi = psi + i chi of a real variable, the derivative of exp(phi) has the same form as for real phi: phi' exp(phi).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\phi^{(n)}(t) = m(m - 1) \dots (m - n + 1)z^{n} (1 + tz)^{m-n}The nth derivative of (1 + tz)^m with respect to t is m(m-1)...(m-n+1) z^n (1 + tz)^(m-n).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\frac{\phi^{n}(0)}{n!} = \binom{m}{n} z^{n}The nth derivative at t = 0, divided by n factorial, gives the binomial coefficient times z^n. (The book writes phi^n(0) for the nth derivative.)
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\phi(1) = \phi(0) + \phi'(0) + \frac{\phi''(0)}{2!} + \dots + \frac{\phi^{(n-1)}(0)}{(n - 1)!} + R_{n}The value of phi at 1 equals its Taylor polynomial of degree n - 1 at 0 plus a remainder R_n.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
R_{n} = \frac{1}{(n - 1)!}\int_{0}^{1} (1 - t)^{n-1} \phi^{(n)}(t)\, dtThe remainder R_n is given by an integral of the nth derivative against (1 - t)^(n-1).
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
|1 + tz| = \sqrtp{1 + 2tr\cos\theta + t^{2}r^{2}} \geq 1 - trFor z = r(cos theta + i sin theta) and 0 <= t <= 1, the modulus |1 + tz| is at least 1 - tr.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
(1 + z)^{m} = \exp\{m\log(1 + z)\}For all real m and real z between -1 and 1, (1 + z)^m equals exp of m times log(1 + z); the general form extends this to complex m and z with |z| < 1, using the principal value of the logarithm.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\left|\frac{a_{n+1}}{a_{n}}\right| = \left|\frac{m - n}{n + 1}\right| \to 1The ratio of successive absolute coefficients of the binomial series tends to 1, whether m is real or complex, so the series converges for |z| < 1.
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS
\frac{d}{dt}(1 + tz)^{m} = mz(1 + tz)^{m-1}Restated for the binomial series argument: derivative of (1 + tz)^m with respect to real t equals m z (1 + tz)^(m-1), each side with its principal value.
Problems
Exercise XCV
Exercise XCV, problem 1, p. 411
Determine the values of $\zeta$ for which $\cos\zeta$ and $\sin\zeta$ are (i) real (ii) purely imaginary. [For example $\cos\zeta$ is real when $\eta = 0$ or when $\xi$ is any multiple of $\pi$.]
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Exercise XCV, problem 10, p. 411
**of $\cos\zeta = \alpha + i\beta$, where $\beta \neq 0$.** We may suppose $\beta > 0$, since the results when $\beta < 0$ may be deduced by merely changing the sign of $i$. In this case = ,0pt minus 3pt= -, (1) and (/)^2 + (/)^2 = 1. If we put $\cosh^{2} \eta = x$ we find that x^2 - (1 + ^2 + ^2)x + ^2 = 0 or $x = (A_{1} ± A_{2})^{2}$, where A_1 = 12(+ 1)^2 + ^2,0pt minus 3ptA_2 = 12(- 1)^2 + ^2. Suppose $\alpha > 0$. Then $A_{1} > A_{2} > 0$ and $\cosh\eta = A_{1} ± A_{2}$. Also = /() = A_1 A_2, and since $\cosh\eta > \cos\xi$ we must take = A_1 + A_2,0pt minus 3pt= A_1 - A_2. The general solutions of these equations are = 2k± M,0pt minus 3pt= ±L + L^2 - 1, (2) where $L = A_{1} + A_{2}$, $M = A_{1} - A_{2}$, and $\arccos M$ lies between $0$ and $\frac{1}{2}\pi$. The values of $\eta$ and $\xi$ thus found above include, however, the solutions of the equations = ,0pt minus 3pt= , (3) as well as those of the equations (1), since we have only used the second of the latter equations after squaring it. To distinguish the two sets of solutions we observe that the sign of $\sin\xi$ is the same as the ambiguous sign in the first of the equations (2), and the sign of $\sinh\eta$ is the same as the ambiguous sign in the second. Since $\beta > 0$, these two signs must be different. Hence the general solution required is = 2k± [M - iL + L^2 - 1].
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Exercise XCV, problem 11, p. 411
Work out the cases in which $\alpha < 0$ and $\alpha = 0$ in the same way.
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Exercise XCV, problem 12, p. 411
If $\beta = 0$ then $L = \frac{1}{2}|\alpha + 1| + \frac{1}{2}|\alpha - 1|$ and $M = \frac{1}{2}|\alpha + 1| - \frac{1}{2}|\alpha - 1|$. Verify that the results thus obtained agree with those of Ex. 8.
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Exercise XCV, problem 13, p. 411
0.375em plus 0.75em minus 0.25emShow that if $\alpha$ and $\beta$ are positive then the general solution of $\sin\zeta = \alpha + i\beta$ is = k+(-1)^k [M + iL + L^2 - 1], where $\arcsin M$ lies between $0$ and $\frac{1}{2}\pi$. Obtain the solution in the other possible cases.
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Exercise XCV, problem 14, p. 411
Solve $\tan\zeta = \alpha$, where $\alpha$ is real. [All the roots are real.]
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Exercise XCV, problem 15, p. 411
Show that the general solution of $\tan \zeta = \alpha + i\beta$, where $\beta \neq 0$, is = k+ 12+ 14 i ^2 + (1 + )^2 ^2 + (1 - )^2 , where $\theta$ is the numerically least angle such that : : 1 :: 1 - ^2 - ^2 : 2: (1 - ^2 - ^2)^2 + 4^2.
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Exercise XCV, problem 16, p. 411
If $z = \xi\exp(\frac{1}{4}\pi i)$, where $\xi$ is real, and $c$ is also real, then the modulus of $\cos 2\pi z - \cos 2\pi c$ is aligned[b] [121 + 4c + (22) &+ (22) &- 42c (2) (2)] aligned.
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Exercise XCV, problem 17, p. 411
Prove that gather* |(+ i)| = (), aligned (+ i) &= () (), (+ i) &= () (). aligned gather*
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Exercise XCV, problem 18, p. 411
Prove that $|\exp\zeta|$ tends to $\infty$ if $\zeta$ moves away towards infinity along any straight line through the origin making an angle less than $\frac{1}{2}\pi$ with $OX$, and to $0$ if $\zeta$ moves away along a similar line making an angle greater than $\frac{1}{2}\pi$ with $OX$.
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Exercise XCV, problem 19, p. 411
Prove that $|\cos\zeta|$ and $|\sin\zeta|$ tend to $\infty$ if $\zeta$ moves away towards infinity along any straight line through the origin other than either half of the real axis.
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Exercise XCV, problem 2, p. 411
alignat*2 |(+ i)| &= ^2 + ^2 &&= 12 (2+ 2), |(+ i)| &= ^2 + ^2 &&= 12 (2- 2). alignat* [Use (*e.g.*) the equation $|\cos(\xi + i\eta)| = \sqrtb{\cos(\xi + i\eta) \cos(\xi - i\eta)}$.]
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Exercise XCV, problem 20, p. 411
Prove that $\tan\zeta$ tends to $-i$ or to $i$ if $\zeta$ moves away to infinity along the straight line of Ex. 19, to $-i$ if the line lies above the real axis and to $i$ if it lies below.
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Exercise XCV, problem 3, p. 411
$\tan (\xi + i \eta) = \dfrac{\sin 2\xi + i\sinh 2\eta}{\cosh 2\eta + \cos 2\xi}$,0pt minus 3pt$\cot (\xi + i \eta) = \dfrac{\sin 2\xi - i\sinh 2\eta}{\cosh 2\eta - \cos 2\xi}$. [For example (+ i) = (+ i) (- i) (+ i) (- i) = 2+ 2i2+ 2i, which leads at once to the result given.]
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Exercise XCV, problem 4, p. 411
align* (+ i ) &= + i 12 (2+ 2), (+ i ) &= - i 12 (2- 2). align*
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Exercise XCV, problem 5, p. 411
If $|\cos (\xi + i\eta)| = 1$ then $\sin^{2} \xi = \sinh^{2} \eta$, and if $|\sin (\xi + i\eta)| = 1$ then $\cos^{2} \xi = \sinh^{2} \eta$.
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Exercise XCV, problem 6, p. 411
If $|\cos (\xi + i\eta)| = 1$, then (+ i) = ±^2 = ±^2 .
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Exercise XCV, problem 7, p. 411
Prove that $\Log \cos (\xi + i\eta) = A + iB$, where A = 12 12 (2+ 2) and $B$ is any angle such that B = -B = 112 (2+ 2). Find a similar formula for $\Log \sin (\xi + i\eta)$.
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Exercise XCV, problem 8, p. 411
**of the equation $\cos\zeta = a$, where $a$ is real.** Putting $\zeta = \xi + i\eta$, and equating real and imaginary parts, we obtain = a,0pt minus 3pt= 0. Hence either $\eta = 0$ or $\xi$ is a multiple of $\pi$. If (i) $\eta = 0$ then $\cos\xi = a$, which is impossible unless $-1 \leq a \leq 1$. This hypothesis leads to the solution = 2k± a, where $\arccos a$ lies between $0$ and $\frac{1}{2}\pi$. If (ii) $\xi = m\pi$ then $\cosh\eta = (-1)^{m}a$, so that either $a \geq 1$ and $m$ is even, or $a \leq -1$ and $m$ is odd. If $a = ± 1$ then $\eta = 0$, and we are led back to our first case. If $|a| > 1$ then $\cosh\eta = |a|$, and we are led to the solutions alignat*4 &=& 2k &± i &&a + a^2 - 10pt minus 3pt&&(a > 1), &=&(2k + 1) &± i-&&a + a^2 - 10pt minus 3pt&&(a < -1). alignat* For example, the general solution of $\cos\zeta = -\frac{5}{3}$ is $\zeta = (2k + 1)\pi ± i\log 3$.
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Exercise XCV, problem 9, p. 411
Solve $\sin\zeta = \alpha$, where $\alpha$ is real.
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Exercise XCVI
Exercise XCVI, problem 1, p. 416
Calculate $\cos i$ and $\sin i$ to two places of decimals by means of the power series for $\cos z$ and $\sin z$.
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Exercise XCVI, problem 10, p. 416
Sum 1 + az1! + a^22z2! + …,0pt minus 3ptaz1! + a^22z2! + ….
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Exercise XCVI, problem 11, p. 416
Sum 1 - 2z2! + 4z4! - …,0pt minus 3ptz1! - 3z3! + … and the corresponding series involving sines.
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Exercise XCVI, problem 12, p. 416
Show that 1 + 4z4! + 8z8! + … = 12(z) (z) + (z) (z).
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Exercise XCVI, problem 13, p. 416
Show that the expansions of $\cos(x + h)$ and $\sin(x + h)$ in powers of $h$ (% [examples:lvi]Ex. lvi%. 1) are valid for all values of $x$ and $h$, real or complex.
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Exercise XCVI, problem 2, p. 416
Prove that $|\cos z| \leq \cosh|z|$ and $|\sin z| \leq \sinh|z|$.
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Exercise XCVI, problem 3, p. 416
Prove that if $|z| < 1$ then $|\cos z| < 2$ and $|\sin z| < \frac{6}{5}|z|$.
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Exercise XCVI, problem 4, p. 416
Since $\sin 2z = 2\sin z \cos z$ we have (2z) - (2z)^33! + (2z)^55! - … = 2(z - z^33! + …) (1 - z^22! + …). Prove by multiplying the two series on the right-hand side ([§]195) and equating coefficients ([§]194) that 2n + 11 + 2n + 13 + …+ 2n + 12n + 1 = 2^2n. Verify the result by means of the binomial theorem. Derive similar identities from the equations ^2z + ^2z = 1,0pt minus 3pt2z = 2^2z - 1 = 1 - 2^2z.
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Exercise XCVI, problem 5, p. 416
Show that (1 + i)z = _0^ 2^12n (14ni) z^nn!.
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Exercise XCVI, problem 6, p. 416
Expand $\\cos z \\cosh z$ in powers of $z$. [We have align* cos z cosh z + isin z sinh z &= cos(1 - i)z = tfrac12 [exp(1 + i)z + exp-(1 + i)z] &= tfrac12 sum_0^infty 2^frac12n 1 + (-1)^n exp(tfrac14npi i) fracz^nn!, align* and similarly cos z cosh z - isin z sinh z = cos (1 + i)z = tfrac12 sum_0^infty 2^frac12n 1 + (-1)^n exp(-tfrac14npi i) fracz^nn!. Hence cos z cosh z = tfrac12 sum_0^infty 2^frac12n1 + (-1)^n cos tfrac14npi fracz^nn! = 1 - frac2^2z^44! + frac2^4z^88! - dots.]
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Exercise XCVI, problem 7, p. 416
Expand $\sin z \sinh z$, $\cos z \sinh z$, and $\sin z \cosh z$ in powers of $z$.
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Exercise XCVI, problem 8, p. 416
Expand $\sin^{2} z$ and $\sin^{3} z$ in powers of $z$. [Use the formulae ^2 z = 12 (1 - 2z),0pt minus 3pt^3 z = 14 (3z - 3z), …. It is clear that the same method may be used to expand $\cos^{n} z$ and $\sin^{n} z$, where $n$ is any integer.]
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Exercise XCVI, problem 9, p. 416
Sum the series C = 1 + z1! + 2z2! + 3z3! +…,0pt minus 3ptS = z1! + 2z2! + 3z3! + …. [Here align* C + iS &= 1 + (iz)1! + (2iz)2! + … = (iz) &= (z) (z) + i(z), align* and similarly C - iS = (-iz) = (z)(z) - i(z). Hence C = (z)(z),0pt minus 3ptS = (z)(z).]
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Exercise XCVII
Exercise XCVII, problem 1, p. 420
Prove that, in any triangle in which $a > b$, c = a - ba C - b^22a^2 2C - …. [Use the formula $\log c = \frac{1}{2} \log(a^{2} + b^{2} - 2ab\cos C )$.]
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Exercise XCVII, problem 2, p. 420
Prove that if $-1 < r < 1$ and $-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi$ then r2 - 12r^2 4 + 13r^3 6- … = - (1 - r1 + r) , the inverse tangent lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$. Determine the sum of the series for all other values of $\theta$.
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Exercise XCVII, problem 3a, p. 420
Prove, by considering the expansions of $\log(1 + iz)$ and $\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$.
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Exercise XCVII, problem 3b, p. 420
Prove, by considering the expansions of $\log(1 + iz)$ and $\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$.
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Exercise XCVII, problem 3c, p. 420
Prove, by considering the expansions of $\log(1 + iz)$ and $\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$.
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Exercise XCVII, problem 3d, p. 420
Prove, by considering the expansions of $\log(1 + iz)$ and $\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$.
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Exercise XCVII, problem 4a, p. 420
Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….
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Exercise XCVII, problem 4b, p. 420
Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….
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Exercise XCVII, problem 4c, p. 420
Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….
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Exercise XCVIII
Exercise XCVIII, problem 1, p. 424
Suppose $m$ real. Then since (1 + z) = 12 (1 + 2r+ r^2) + i(r1 + r), we obtain align* _0^ mn z^n &= 12m (1 + 2r+ r^2) m(r1 + r) &= (1 + 2r+ r^2)^12m m(r1 + r), align* all the inverse tangents lying between $-\frac{1}{2}\pi$ and $\frac{1}{2}\pi$. In particular, if we suppose $\theta = \frac{1}{2}\pi$, $z = ir$, and equate the real and imaginary parts, we obtain align* 1 - m2 r^2 + m4 r^4 - … &= (1 + r^2)^12m (mr), m1 r - m3 r^3 + m5 r^5 - … &= (1 + r^2)^12m (mr). align*
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Exercise XCVIII, problem 2, p. 424
Verify the formulae of Ex. 1 when $m = 1$, $2$, $3$. [Of course when $m$ is a positive integer the series is finite.]
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Exercise XCVIII, problem 3, p. 424
Prove that if $0 \leq r < 1$ then align* 1 - 1·32·4 r^2 + 1·3·5·72·4·6·8 r^4 - … &= 1 + r^2 + 12(1 + r^2), 12 r - 1·3·52·4·6 r^3 + 1·3·5·7·92·4·6·8·10 r^5 - … &= 1 + r^2 - 12(1 + r^2). align* [Take $m = -\frac{1}{2}$ in the last two formulae of Ex. 1.]
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Exercise XCVIII, problem 4, p. 424
Prove that if $-\frac{1}{4}\pi < \theta < \frac{1}{4}\pi$ then align* m&= ^m 1 - m2 ^2 + m4 ^4 - …, m&= ^m m1 - m3 ^3 + …, align* for all real values of $m$. [These results follow at once from the equations m+ im = (+ i)^m = ^m (1 + i)^m.]
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Exercise XCVIII, problem 5, p. 424
We proved (% [examples:lxxxi]Ex. lxxxi%. 6), by direct multiplication of series, that $f(m, z) = \sum\dbinom{m}{n} z^{n}$, where $|z| < 1$, satisfies the functional equation f(m, z) f(m’, z) = f(m + m’, z). Deduce, by an argument similar to that of [§]216, and without assuming the general result of p.423, that if $m$ is real and rational then f(m, z) = m(1 + z).
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Exercise XCVIII, problem 6, p. 424
If $z$ and $\mu$ are real, and $-1 < z < 1$, then in z^n = (1 + z) + i(1 + z).
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Exercise Misc-X
Exercise Misc-X, problem 1, p. 425
Show that the real part of $i^{\log(1+i)}$ is e^(4k+1)^2/8 14(4k + 1)2, where $k$ is any integer.
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Exercise Misc-X, problem 10, p. 425
The equation $\tan z = a\tanh cz$, where $a$ and $c$ are real, has an infinity of real and of purely imaginary roots, but no complex roots.
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Exercise Misc-X, problem 11, p. 425
Show that if $x$ is real then e^ax bx = _0^ x^nn! a^n - n2 a^n-2 b^2 + n4 a^n-4 b^4 - …, where there are $\frac{1}{2}(n + 1)$ or $\frac{1}{2}(n + 2)$ terms inside the large brackets. Find a similar series for $e^{ax} \sin bx$.
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Exercise Misc-X, problem 12, p. 425
If $n\phi(z, n) \to z$ as $n \to \infty$, then $\{1 + \phi(z, n)\}^{n} \to \exp z$.
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Exercise Misc-X, problem 13, p. 425
If $\phi(t)$ is a complex function of the real variable $t$, then ddt (t) = ’(t)(t). %[** TN: Paragraph break added] [Use the formulae = + i,0pt minus 3pt= 12(^2 + ^2) + i(/).]
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Exercise Misc-X, problem 14, p. 425
**.** In Ch.III (xxi. 21 *et seq.*, and [misc:III]Misc. Exs. 22 *et seq.*) we considered some simple examples of the geometrical relations between figures in the planes of two variables $z$, $Z$ connected by a relation $z = f(Z)$. We shall now consider some cases in which the relation involves logarithmic, exponential, or circular functions. Suppose firstly that z = (Z/a),0pt minus 3ptZ = (a/) z where $a$ is positive. To one value of $Z$ corresponds one of $z$, but to one of $z$ infinitely many of $Z$. If $x$, $y$, $r$, $\theta$ are the coordinates of $z$ and $X$, $Y$, $R$, $\Theta$ those of $Z$, we have the relations alignat*2 x &= e^X/a (Y/a), & y &= e^X/a (Y/a), X &= (a/) r, & Y &= (a/) + 2ka, alignat* where $k$ is any integer. If we suppose that $-\pi < \theta \leq \pi$, and that $\Log z$ has its principal value $\log z$, then $k = 0$, and $Z$ is confined to a strip of its plane parallel to the axis $OX$ and extending to a distance $a$ from it on each side, one point [pg]427 of this strip corresponding to one of the whole $z$-plane, and conversely. By taking a value of $\Log z$ other than the principal value we obtain a similar relation between the $z$-plane and another strip of breadth $2a$ in the $Z$-plane. To the lines in the $Z$-plane for which $X$ and $Y$ are constant correspond the circles and radii vectores in the $z$-plane for which $r$ and $\theta$ are constant. To one of the latter lines corresponds the whole of a parallel to $OX$, but to a circle for which $r$ is constant corresponds only a part, of length $2a$, of a parallel to $OY$. To make $Z$ describe the whole of the latter line we must make $z$ move continually round and round the circle.
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Exercise Misc-X, problem 15, p. 425
Show that to a straight line in the $Z$-plane corresponds an equiangular spiral in the $z$-plane.
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Exercise Misc-X, problem 16, p. 425
Discuss similarly the transformation $z = c\cosh(\pi Z/a)$, showing in particular that the whole $z$-plane corresponds to any one of an infinite number of strips in the $Z$-plane, each parallel to the axis $OX$ and of breadth $2a$. Show also that to the line $X = X_{0}$ corresponds the ellipse xc(X_0/a)^2 + yc(X_0/a)^2 = 1, and that for different values of $X_{0}$ these ellipses form a confocal system; and that the lines $Y = Y_{0}$ correspond to the associated system of confocal hyperbolas. Trace the variation of $z$ as $Z$ describes the whole of a line $X = X_{0}$ or $Y = Y_{0}$. How does $Z$ vary as $z$ describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of $x$?
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Exercise Misc-X, problem 17, p. 425
Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch.III, [misc:III]Misc. Ex. 25. [The transformation $z = c\cosh(\pi Z/a)$ may be regarded as compounded from the transformations z = cz_1,0pt minus 3ptz_1 = 12z_2 + (1/z_2),0pt minus 3ptz_2 = (Z/a).]
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Exercise Misc-X, problem 18, p. 425
Discuss similarly the transformation $z = c\tanh(\pi Z/a)$, showing that to the lines $X = X_{0}$ correspond the coaxal circles x - c(2X_0/a)^2 + y^2 = c^2^2(2X_0/a), and to the lines $Y = Y_{0}$ the orthogonal system of coaxal circles.
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Exercise Misc-X, problem 19, p. 425
**Stereographic and Mercator’s Projections.** The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are $0$, $0$, $-1$) on to the tangent plane at the north pole. The coordinates of a point on the sphere are $\xi$, $\eta$, $\zeta$, and Cartesian axes $OX$, $OY$ are taken on the tangent plane, parallel to the axes of $\xi$ and $\eta$. Show that the coordinates of the projection of the point are x = 2/(1 + ),0pt minus 3pty = 2/(1 + ), and that $x + iy = 2\tan \frac{1}{2}\theta \Cis\phi$, where $\phi$ is the longitude (measured from the plane $\eta = 0$) and $\theta$ the north polar distance of the point on the sphere. [pg]428 This projection gives a map of the sphere on the tangent plane, generally known as the *Stereographic Projection*. If now we introduce a new complex variable Z = X + iY = -i12z = -i12(x + iy) so that $X = \phi$, $Y = \log \cot \frac{1}{2}\theta$, we obtain another map in the plane of $Z$, usually called *Mercator’s Projection*. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.
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Exercise Misc-X, problem 2, p. 425
If $a\cos\theta + b\sin\theta + c = 0$, where $a$, $b$, $c$ are real and $c^{2} > a^{2} + b^{2}$, then = m+ ± i|c| + c^2 - a^2 - b^2a^2 + b^2, where $m$ is any odd or any even integer, according as $c$ is positive or negative, and $\alpha$ is an angle whose cosine and sine are $a/\sqrtp{a^{2} + b^{2}}$ and $b/\sqrtp{a^{2} + b^{2}}$.
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Exercise Misc-X, problem 20, p. 425
Discuss the transformation given by the equation z = (Z - aZ - b), showing that the straight lines for which $x$ and $y$ are constant correspond to two orthogonal systems of coaxal circles in the $Z$-plane.
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Exercise Misc-X, problem 21, p. 425
Discuss the transformation z = Z - a + Z - bb - a, showing that the straight lines for which $x$ and $y$ are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points $Z = a$ and $Z = b$. [We have alignat*2 Z - a + Z - b &= b - a (& &x + iy), Z - a - Z - b &= b - a (&-&x - iy); alignat* and it will be found that |Z - a| + |Z - b| = |b - a|2x,0pt minus 3pt|Z - a| - |Z - b| = |b - a|2y.]
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Exercise Misc-X, problem 22, p. 425
**transformation $z = Z^{i}$.** If $z = Z^{i}$, where the imaginary power has its principal value, we have (r + i) = z = (iZ) = (iR - ), so that $\log r = -\Theta$, $\theta = \log R + 2k\pi$, where $k$ is an integer. As all values of $k$ give the same point $z$, we shall suppose that $k = 0$, so that r = -,0pt minus 3pt= R. (1) The whole plane of $Z$ is covered when $R$ varies through all positive values and $\Theta$ from $-\pi$ to $\pi$: then $r$ has the range $\exp(-\pi)$ to $\exp\pi$ and $\theta$ ranges through all real values. Thus the $Z$-plane corresponds to the ring bounded by the circles $r = \exp(-\pi)$, $r = \exp\pi$; but this ring is covered infinitely often. If however $\theta$ is allowed to vary only between $-\pi$ and $\pi$, so that the ring is covered only once, then $R$ can vary only from $\exp(-\pi)$ to $\exp \pi$, so that the variation of $Z$ is restricted to a ring similar in all respects to that within which $z$ varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which $z$ (or $Z$) must not cross, as its amplitude must not transgress the limits $-\pi$ and $\pi$. [pg]429 We thus obtain a correspondence between two rings, given by the pair of equations z = Z^i,0pt minus 3ptZ = z^-i, where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.
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Exercise Misc-X, problem 23, p. 425
Trace the variation of $z$ when $Z$, starting at the point $\exp \pi$, moves round the larger circle in the positive direction to the point $-\exp \pi$, along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position.
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Exercise Misc-X, problem 24, p. 425
Suppose each plane to be divided up into an infinite series of rings by circles of radii …,0pt minus 3pte^-(2n+1), …,0pt minus 3pte^-,0pt minus 3pte^,0pt minus 3pte^3, …,0pt minus 3pte^(2n+1), …. Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations $z = Z^{i}$, $Z = z^{-i}$.
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Exercise Misc-X, problem 25, p. 425
If $z = Z^{i}$, any value of the power being taken, and $Z$ moves along an equiangular spiral whose pole is the origin in its plane, then $z$ moves along an equiangular spiral whose pole is the origin in its plane.
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Exercise Misc-X, problem 26, p. 425
How does $Z = z^{ai}$, where $a$ is real, behave as $z$ approaches the origin along the real axis. [$Z$ moves round and round a circle whose centre is the origin (the unit circle if $z^{ai}$ has its principal value), and the real and imaginary parts of $Z$ both oscillate finitely.]
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Exercise Misc-X, problem 27, p. 425
Discuss the same question for $Z = z^{a+bi}$, where $a$ and $b$ are any real numbers.
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Exercise Misc-X, problem 28, p. 425
Show that the region of convergence of a series of the type $\sum\limits_{-\infty}^{\infty} a_{n}z^{nai}$, where $a$ is real, is an angle, *i.e.* a region bounded by inequalities of the type $\theta_{0} < \am z < \theta_{1}$ [The angle may reduce to a line, or cover the whole plane.]
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Exercise Misc-X, problem 29, p. 425
**Curves.** If $f(z)$ is a function of the complex variable $z$, we call the curves for which $|f(z)|$ is constant the *level curves* of $f(z)$. Sketch the forms of the level curves of alignat*2 z - a 0pt minus 3pt& (*concentric circles*), & (z - a)(z - b) 0pt minus 3pt& (*Cartesian ovals*), (z - a)/(z - b) 0pt minus 3pt& (*coaxal circles*), & z 0pt minus 3pt& (*straight lines*). alignat*
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Exercise Misc-X, problem 3, p. 425
Prove that if $\theta$ is real and $\sin\theta \sin\phi = 1$ then = (k + 12)± i12(k+ ), where $k$ is any even or any odd integer, according as $\sin\theta$ is positive or negative.
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Exercise Misc-X, problem 30, p. 425
Sketch the forms of the level curves of $(z - a)(z - b)(z - c)$, $(1 + z\sqrt{3} + z^{2})/z$. [Some of the level curves of the latter function are drawn in [fig:59]Fig. 59, the curves marked i--vii corresponding to the values .10,0pt minus 3pt2 - 3 = .27,0pt minus 3pt.40,0pt minus 3pt1.00,0pt minus 3pt2.00,0pt minus 3pt2 + 3 = 3.73,0pt minus 3pt4.53 of $|f(z)|$. The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.]
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Exercise Misc-X, problem 31i, p. 425
Sketch the forms of the level curves of (i) $z\exp z$, (ii) $\sin z$. [See [fig:60]Fig. 60, which represents the level curves of $\sin z$. The curves marked i--viii correspond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$, $4.00$.]
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Exercise Misc-X, problem 31ii, p. 425
Sketch the forms of the level curves of (i) $z\exp z$, (ii) $\sin z$. [See [fig:60]Fig. 60, which represents the level curves of $\sin z$. The curves marked i--viii correspond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$, $4.00$.]
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Exercise Misc-X, problem 32, p. 425
Sketch the forms of the level curves of $\exp z - c$, where $c$ is a real constant. [[fig:61]Fig. 61 shows the level curves of $|\exp z - 1|$, the curves i--vii corresponding to the values of $k$ given by $\log k = -1.00$, $-.20$, $-.05$, $0.00$, $.05$, $.20$, $1.00$.]
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Exercise Misc-X, problem 33, p. 425
The level curves of $\sin z - c$, where $c$ is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether $c < 1$ or $c > 1$. In [fig:62]Fig. 62 we have taken $c = .5$, and the curves i--viii correspond to $k = .29$, $.37$, $.50$, $.87$, $1.50$, $2.60$, $4.50$, $7.79$. In [fig:63]Fig. 63 we have taken $c = 2$, and the curves i--vii correspond to $k = .58$, $1.00$, $1.73$, $3.00$, $5.20$, $9.00$, $15.59$. If $c = 1$ then the curves are the same as those of [fig:60]Fig. 60, except that the origin and scale are different.]
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Exercise Misc-X, problem 34, p. 425
Prove that if $0 < \theta < \pi$ then alignat*3 &+ 13 3&&+ 15 5&&+ … = 14 ^212, &+ 13 3&&+ 15 5&&+ … = 14, alignat* and determine the sums of the series for all other values of $\theta$ for which they are convergent. [Use the equation z + 13z^3 + 15z^5 + … = 12 (1 + z1 - z) where $z = \cos\theta + i\sin\theta$. When $\theta$ is increased by $\pi$ the sum of each series simply changes its sign. It follows that the first formula holds for all values of $\theta$ save multiples of $\pi$ (for which the series diverges), while the sum of the second series is $\frac{1}{4}\pi$ if $2k\pi < \theta < (2k + 1)\pi$, $-\frac{1}{4}\pi$ if $(2k + 1)\pi < \theta < (2k + 2)\pi$, and $0$ if $\theta$ is a multiple of $\pi$.]
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Exercise Misc-X, problem 35, p. 425
Prove that if $0 < \theta < \frac{1}{2}\pi$ then alignat*3 &- 13 3&&+ 15 5&&- … = 14, &- 13 3&&+ 15 5&&- … = 14 (+ )^2; alignat* and determine the sums of the series for all other values of $\theta$ for which they are convergent.
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Exercise Misc-X, problem 36, p. 425
Prove that + 12 22 + 13 33+ … = -14 4(- )^2, unless $\theta - \alpha$ or $\theta + \alpha$ is a multiple of $2\pi$.
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Exercise Misc-X, problem 37, p. 425
Prove that if neither $a$ nor $b$ is real then _0^ dx(x - a)(x - b) = -(-a) - (-b)a - b, each logarithm having its principal value. Verify the result when $a = ci$, $b = -ci$, where $c$ is positive. Discuss also the cases in which $a$ or $b$ or both are real and negative.
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Exercise Misc-X, problem 38, p. 425
Prove that if $\alpha$ and $\beta$ are real, and $\beta > 0$, then _0^ dx^2 - (+ i)^2 = i2(+ i). What is the value of the integral when $\beta < 0$?
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Exercise Misc-X, problem 39, p. 425
Prove that, if the roots of $Ax^{2} + 2Bx + C = 0$ have their imaginary parts of opposite signs, then _-^ dxAx^2 + 2Bx + C = iB^2 - AC, the sign of $\sqrtp{B^{2} - AC}$ being so chosen that the real part of $\{\sqrtp{B^{2} - AC}\}/Ai$ is positive.
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Exercise Misc-X, problem 4, p. 425
Show that if $x$ is real then gather* ddx (a + ib)x = (a + ib) (a + ib) x, (a + ib)x dx = (a + ib)xa + ib. gather* Deduce the results of % [examples:lxxxvii]Ex. lxxxvii%. 3.
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Exercise Misc-X, problem 5, p. 425
Show that if $a > 0$ then $\ds\int_{0}^{\infty} \exp\{-(a + ib)x\}\, dx = \frac{1}{a + ib}$, and deduce the results of % [examples:lxxxvii]Ex. lxxxvii%. 5.
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Exercise Misc-X, problem 6, p. 425
Show that if $(x/a)^{2} + (y/b)^{2} = 1$ is the equation of an ellipse, and $f(x, y)$ denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of $2\pi$ from -if(a, ib) - f(a, -ib). [The eccentric angles are given by $f(a\cos\alpha, b\sin\alpha) + \dots = 0$ or by f12 a (u + 1u), -12 ib (u - 1u) + …= 0, where $u = \exp i\alpha$; and $\sum\alpha$ is equal to one of the values of $-i\Log P$, where $P$ is the product of the roots of this equation.]
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Exercise Misc-X, problem 7, p. 425
Determine the number and approximate positions of the roots of the equation $\tan z = az$, where $a$ is real. [We know already (% [examples:xvii]Ex. xvii%. 4) that the equation has infinitely many real roots. Now let $z = x + iy$, and equate real and imaginary parts. We obtain 2x/(2x + 2y) = ax,0pt minus 3pt2y/(2x + 2y) = ay, so that, unless $x$ or $y$ is zero, we have (2x)/2x = (2y)/2y. [pg]426 This is impossible, the left-hand side being numerically less, and the right-hand side numerically greater than unity. Thus $x = 0$ or $y = 0$. If $y = 0$ we come back to the real roots of the equation. If $x = 0$ then $\tanh y = ay$. It is easy to see that this equation has no real root other than zero if $a \leq 0$ or $a \geq 1$, and two such roots if $0 < a < 1$. Thus there are two purely imaginary roots if $0 < a < 1$; otherwise all the roots are real.]
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Exercise Misc-X, problem 8, p. 425
The equation $\tan z = az + b$, where $a$ and $b$ are real and $b$ is not equal to zero, has no complex roots if $a \leq 0$. If $a > 0$ then the real parts of all the complex roots are numerically greater than $|b/2a|$.
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Exercise Misc-X, problem 9, p. 425
The equation $\tan z = a/z$, where $a$ is real, has no complex roots, but has two purely imaginary roots if $a < 0$.
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Exercise XCIII
Exercise XCIII, problem 1, p. 401
We supposed above that $-\pi < \theta < \pi$, and so excluded the case in which $z$ is *real and negative*. In this case the straight line from $1$ to $z$ passes through $0$, and is therefore not admissible as a path of integration. Both $\pi$ and $-\pi$ are values of $\am z$, and $\theta$ is equal to one or other of them: also $r = -z$. The values of $\Log z$ are still the values of $\log |z| + i\am z$, viz. (-z) + (2k + 1)i, where $k$ is an integer. The values $\log (-z) + \pi i$ and $\log (-z) - \pi i$ correspond to paths from $1$ to $z$ lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of $\Log z$, as convenience dictates. We shall choose the value $\log (-z) + \pi$ i corresponding to the first path.
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Exercise XCIII, problem 10, p. 401
The function $f(x)$ defined by f(x) = p+ (q - p)(x - 1) + (r - q)(x) is equal to $p$ when $x > 1$, to $q$ when $0 < x < 1$, and to $r$ when $x < 0$.
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Exercise XCIII, problem 11, p. 401
For what values of $z$ is (i) $\log z$ (ii) any value of $\Log z$ (*a*) real or (*b*) purely imaginary?
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Exercise XCIII, problem 12, p. 401
If $z = x + iy$ then $\Log\Log z = \log R + i(\Theta + 2k'\pi)$, where R^2 = (r)^2 + (+ 2k)^2 and $\Theta$ is the least positive angle determined by the equations : : 1 :: r : + 2k: (r)^2 + (+ 2k)^2. Plot roughly the doubly infinite set of values of $\Log\Log(1 + i\sqrt{3})$, indicating which of them are values of $\log\Log(1 + i \sqrt{3})$ and which of $\Log\log(1 + i\sqrt{3})$.
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Exercise XCIII, problem 2, p. 401
The real and imaginary parts of any value of $\Log z$ are both continuous functions of $x$ and $y$, except for $x = 0$, $y = 0$.
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Exercise XCIII, problem 3, p. 401
**functional equation satisfied by $\Log z$.** The function $\Log z$ satisfies the equation z_1 z_2 = z_1 + z_2, (1) in the sense that *every* value of either side of this equation is *one* of the values of the other side. This follows at once by putting z_1 = r_1(_1 + i_1),0pt minus 3ptz_2 = r_2(_2 + i_2), and applying the formula of p.401. It is however not true that z_1z_2 = z_1 + z_2 (2) in all circumstances. If, *e.g.*, z_1 = z_2 = 12(-1 + i3) = 23+ i 23, then $\log z_{1} = \log z_{2} = \frac{2}{3}\pi i$, and $\log z_{1} + \log z_{2} = \frac{4}{3}\pi i$, which is one of the values of $\Log z_{1}z_{2}$, but not the principal value. In fact $\log z_{1}z_{2} = -\frac{2}{3}\pi i$. An equation such as (1), in which every value of either side is a value of the other, we shall call a *complete* equation, or an equation which is *completely true*.
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Exercise XCIII, problem 4, p. 401
The equation $\Log z^{m} = m\Log z$, where $m$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.
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Exercise XCIII, problem 5, p. 401
The equation $\Log (1/z) = -\Log z$ is completely true. It is also true that $\log (1/z) = -\log z$, except when $z$ is real and negative.
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Exercise XCIII, problem 6, p. 401
The equation (z - az - b) = (z - a) - (z - b) is true if $z$ lies outside the region bounded by the line joining the points $z = a$, $z = b$, and lines through these points parallel to $OX$ and extending to infinity in the negative direction.
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Exercise XCIII, problem 7, p. 401
The equation (a - zb - z) = (1 - az) - (1 - bz) is true if $z$ lies outside the triangle formed by the three points $O$, $a$, $b$.
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Exercise XCIII, problem 8, p. 401
Draw the graph of the function $\Imag(\Log x)$ of the real variable $x$. [The graph consists of the positive halves of the lines $y = 2k\pi$ and the negative halves of the lines $y = (2k + 1)\pi$.]
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Exercise XCIII, problem 9, p. 401
The function $f(x)$ of the real variable $x$, defined by f(x) = p+ (q - p)(x), is equal to $p$ when $x$ is positive and to $q$ when $x$ is negative.
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Exercise XCIV
Exercise XCIV, problem 1, p. 407
Find all the values of $i^{i}$. [By definition i^i = (ii). But i = 12+ i12,0pt minus 3pti = (2k + 12)i, where $k$ is any integer. Hence i^i = -(2k + 12) = e^-(2k + 12). All the values of $i^{i}$ are therefore real and positive.]
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Exercise XCIV, problem 10, p. 407
For what values of $\zeta$ is (*a*) any value (*b*) the principal value of $e^{\zeta}$ (i) real (ii) purely imaginary (iii) of unit modulus?
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Exercise XCIV, problem 11, p. 407
The necessary and sufficient conditions that all the values of $a^{\zeta}$ should be real are that $2\xi$ and $\{\eta\log |a| + \xi\am a\}/\pi$, where $\am a$ denotes any value of the amplitude, should both be integral. What are the corresponding conditions that all the values should be of unit modulus?
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Exercise XCIV, problem 12, p. 407
The general value of $|x^{i} + x^{-i}|$, where $x > 0$, is e^-(m-n) 22(m + n)+ (2x).
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Exercise XCIV, problem 13, p. 407
Explain the fallacy in the following argument: since $e^{2m\pi i} = e^{2n\pi i} = 1$, where $m$ and $n$ are any integers, therefore, raising each side to the power $i$ we obtain $e^{-2m\pi} = e^{-2n\pi}$.
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Exercise XCIV, problem 14, p. 407
In what circumstances are any of the values of $x^{x}$, where $x$ is real, themselves real? [If $x > 0$ then x^x = (xx) = (xx) 2mx, the first factor being real. The principal value, for which $m = 0$, is always real. If $x$ is a rational fraction $p/(2q + 1)$, or is irrational, then there is no other real value. But if $x$ is of the form $p/2q$, then there is one other real value, viz. $-\exp (x\log x)$, given by $m = q$. If $x = -\xi < 0$ then x^x = -(-) = (-) -(2m + 1). The only case in which any value is real is that in which $\xi = p/(2q + 1)$, when $m = q$ gives the real value (-) (-p) = (-1)^p ^-. The cases of reality are illustrated by the examples (13)^1/3 = [3]13,0pt minus 3pt(12)^12 = ±12,0pt minus 3pt(-23)^-23 = [3]94,0pt minus 3pt(-13)^-13 = -[3]3.]
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Exercise XCIV, problem 15, p. 407
**to any base.** We may define $\zeta = \Log_{a} z$ in two different ways. We may say (i) that $\zeta = \Log_{a} z$ if the *principal* value of $a^{\zeta}$ is equal to $z$; or we may say (ii) that $\zeta = \Log_{a} z$ if *any* value of $a^{\zeta}$ is equal to $z$. Thus if $a = e$ then $\zeta = \Log_{e} z$, according to the first definition, if the principal value of $e^{\zeta}$ is equal to $z$, or if $\exp \zeta = z$; and so $\Log_{e} z$ is identical with $\Log z$. But, according to the second definition, $\zeta = \Log_{e} z$ if e^ = (e) = z,0pt minus 3pte = z, or $\zeta = (\Log z)/(\Log e)$, any values of the logarithms being taken. Thus = _e z = |z| + (z + 2m)i1 + 2ni, so that $\zeta$ is a doubly infinitely many-valued function of $z$. And generally, according to this definition, $\Log_{a} z = (\Log z)/(\Log a)$.
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Exercise XCIV, problem 16, p. 407
$\Log_{e} 1 = 2m\pi i/(1 + 2n\pi i)$, $\Log_{e}(-1) = (2m + 1)\pi i/(1 + 2n\pi i)$, where $m$ and $n$ are any integers.
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Exercise XCIV, problem 2, p. 407
Find all the values of $(1 + i)^{i}$, $i^{1+i}$, $(1 + i)^{1+i}$.
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Exercise XCIV, problem 3, p. 407
The values of $a^{\zeta}$, when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of $a$. % [0]% (*Math. Trip.* 1899.)% [1]% [If $a^{\zeta} = r(\cos\theta + i\sin\theta)$ we have r = e^- (+ 2m),0pt minus 3pt= + (+ 2m); and all the points lie on the spiral $r = \sigma^{(\xi^{2} + \eta^{2})/\xi} e^{-\eta \theta/\xi}$.]
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Exercise XCIV, problem 4, p. 407
**function $e^{\zeta}$.** If we write $e$ for $a$ in the general formula, so that $\log \sigma = 1$, $\psi = 0$, we obtain e^ = e^-2m (+ 2m) + i(+ 2m). The principal value of $e^{\zeta}$ is $e^{\xi}(\cos\eta + i\sin\eta)$, which is equal to $\exp \zeta$ ([§]223). In particular, if $\zeta$ is real, so that $\eta = 0$, we obtain e^ (2m+ i2m) as the general and $e^{\zeta}$ as the principal value, $e^{\zeta}$ denoting here the positive value of the exponential defined in Ch.IX.
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Exercise XCIV, problem 5, p. 407
Show that $\Log e^{\zeta} = (1 + 2m\pi i)\zeta + 2n\pi i$, where $m$ and $n$ are any integers, and that in general $\Log a^{\zeta}$ has a double infinity of values.
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Exercise XCIV, problem 6, p. 407
The equation $1/a^{\zeta} = a^{-\zeta}$ is completely true (% [examples:xciii]Ex. xciii%. 3): it is also true of the principal values.
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Exercise XCIV, problem 7, p. 407
The equation $a^{\zeta} × b^{\zeta} = (ab)^{\zeta}$ is completely true but not always true of the principal values.
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Exercise XCIV, problem 8, p. 407
The equation $a^{\zeta} × a^{\zeta'} = a^{\zeta+\zeta'}$ is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left-hand side, but the general value of $a^{\zeta} × a^{\zeta'}$, viz. (a + 2mi) + ’(a + 2ni), is not as a rule a value of $a^{\zeta+\zeta'}$ unless $m = n$.]
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Exercise XCIV, problem 9, p. 407
What are the corresponding results as regards the equations a^ = a,0pt minus 3pt(a^)^’ = (a^’)^ = a^’?
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