Appendix

Excerpts

• Appendix

  has a complex real or imaginary root.

• Appendix

  This becomes intuitive geometrically. The portion of the graph of $y = F(x)$ which extends from its point with the abscissa $2$ to its point with the abscissa $3$ either has a lowest point or else has several equally low points, each lower than all the remaining points.

• Appendix

  This simplified proof consists in showing that the two curves represented by $\phi(x, y) = 0$ and $\psi(x, y) = 0$ have at least one point $(x_1, y_1)$ in common, so that $z_1 = x_1 + iy_1$ is a root of $f(z)= 0$.

• Appendix

  In other words, if $|f(z)|\leqq P$, the point representing $z$ is inside circle $C$.

• Appendix

  The proof differs from that of the auxiliary theorem in §62 only in reading “in absolute value” for “numerically.”

Equations

• Appendix

  f(z) \equiv z^n + a_1 z^{n-1} + \dotsb + a_n = 0

  The polynomial equation of degree n with complex coefficients, which the theorem says always has a complex root.

• Appendix

  f(z) = \phi(x,y) + i\psi(x,y)

  Splits the polynomial f(z) into a real part phi and an imaginary part psi, both real polynomials in x and y.

• Appendix

  f(z) = z^n(1+D)

  Factors the polynomial as z to the power n times one plus a remainder term D that is small when z is large.

• Appendix

  D \equiv a_1\left(\frac{1}{z}\right) + \dotsb + a_n\left(\frac{1}{z}\right)^n

  Defines D as the sum of the coefficients a_1 to a_n times powers of 1/z.

• Appendix

  |f(z)| \geqq |z|^n \bigl[1 - |D|\bigr]

  Lower bound on the absolute value of f(z) obtained from the triangle inequality for absolute values.

• Appendix

  |f(z)| > \rho^n(1-p) \geqq P

  Shows that |f(z)| exceeds any preassigned P once rho is large enough.

• Appendix

  \rho \geqq \sqrt[n]{\frac{P}{1-p}} \equiv R

  Defines the radius R so that |z| at least R guarantees |f(z)| greater than P.

• Appendix

  f(a+h) = f(a) + f'(a)h + \dotsb + f^{(r)}(a)·\frac{h^r}{r!} + \dotsb + f^{(n)}(a)·\frac{h^n}{n!}

  Expands f(a+h) in powers of h, with derivatives of f at a as coefficients.

• Appendix

  g(h) \equiv 1 + bh^r + ch^{r+1} + \dotsb + lh^n

  Defines the simplified function g(h) as the normalized expansion of f(a+h) divided by f(a).

• Appendix

  h = \rho(\cos \theta + i \sin \theta)

  Writes the complex increment h in trigonometric form with modulus rho and argument theta.

• Appendix

  b = |b|(\cos \beta + i \sin \beta)

  Writes the complex coefficient b in trigonometric form with modulus |b| and argument beta.

• Appendix

  bh^r = |b| \rho^r \bigl\{\cos(\beta+r\theta) + i\sin (\beta+r\theta)\bigr\}

  The product b h^r in trigonometric form, with argument beta plus r theta, by de Moivre's rule.

• Appendix

  g(h) = (1 - |b|\rho^r) + h^r(ch + \dotsb + lh^{n-r})

  Rewrites g(h) so that its size is controlled by the first term 1 minus |b| rho^r.

• Appendix

  |ch + \dotsb + lh^{n-r}| < |b|

  Chooses rho small enough that the remaining terms of g(h) are smaller in absolute value than |b|.

• Appendix

  |b| \rho^r < 1

  Chooses rho small enough that the product |b| rho^r is less than 1.

• Appendix

  |g(h)| < (1 - |b|\rho^r) + \rho^r|b|

  Bounds |g(h)| by a quantity that is less than 1, so |f(a+h)/f(a)| is less than 1.

• Appendix

  G(x,y) = \phi^2(x, y) + \psi^2(x, y)

  Defines G as the sum of the squares of the real and imaginary parts of f, so that G is the square of |f(z)|.

• Appendix

  x^2 + y^2 = R^2

  The circle C of radius R in the plane of the real and imaginary parts of z.

• Appendix

  G(x_1,y_1) \leqq G(x,y)

  The value G(x_1, y_1) is a minimum of G over the closed disc x^2 + y^2 at most R^2.

• Appendix

  |f(z)|^2 = G(x,y)

  The square of the absolute value of f(z) equals G(x, y).

• Appendix

  |f(z_1)|\leqq |f(z)|

  The absolute value of f is smallest at z_1 among all z on or within the circle C.

• Appendix

  |f(z_1)|\leqq |f(z')| < P

  The minimum value of |f| on the circle C is at most |f(z')| and therefore less than P.

• Appendix

  |f(z)| < |f(z_1)|

  If f(z_1) were nonzero, some z would give a smaller |f(z)| than |f(z_1)|, which contradicts the minimum.

• Appendix

  f^{(n)}(a) = n!

  The n-th derivative of f is the constant n factorial, because the leading coefficient of f is 1.

• Appendix

  z = x+iy

  Writes the complex variable z in terms of its real part x and imaginary part y.

• Appendix

  a_1 = c_1 + id_1

  Writes the complex coefficient a_1 with real part c_1 and imaginary part d_1.

• Appendix

  \rho\equiv |z|

  Defines rho as the absolute value of z.

Problems

No exercises in this chapter.