The Proof that every Equation has a Root

Excerpts

• The Proof that every Equation has a Root

  We can represent the values of $z$ and $Z$ by points in two planes, which we may call the $z$-plane and the $Z$-plane respectively. It is evident that if $z$ describes a closed path $\gamma$ in the $z$-plane, then $Z$ describes a corresponding closed path $\Gamma$ in the $Z$-plane.

• The Proof that every Equation has a Root

  Thus $\am Z$ denotes a one-valued and continuous function of $X$ and $Y$, the real and imaginary parts of $Z$.

• The Proof that every Equation has a Root

  when $z$ describes any contour $\gamma$ in the positive sense the increment of $\am Z$ is $2k\pi$, where $k$ is the number of roots of $Z = 0$ inside $\gamma$, multiple roots being counted multiply.

• The Proof that every Equation has a Root

  Show that the roots of $f'(z) = 0$ are the foci of the ellipse which touches the sides of the triangle $(z_{1}, z_{2}, z_{3})$ at their middle points.

• The Proof that every Equation has a Root

  Thus if its path is like (*b*) in [fig:B]Fig. B, winding once round the origin in the positive direction, then its amplitude will have increased by $2\pi$.

• The Proof that every Equation has a Root

  Thus $PQ$ will have been described twice, once from $P$ to $Q$ and once from $Q$ to $P$. As $z$ moves from $P$ to $Q$, $\am Z$ varies continuously, since $Z$ does not pass through the origin; and if the increment of $\am Z$ is in this case $\theta$, then its increment when $z$ moves from $Q$ to $P$ is $-\theta$; so that, when we add up the increments of $\am Z$ due to the description of the various parts of the smaller contours, all cancel one another, save the increments due to the description of parts of $\gamma$ itself.

• The Proof that every Equation has a Root

  Hence, if $\am Z$ is changed when $z$ describes $\gamma$, there must be *at least one* of the smaller contours, say $\gamma_{1}$, such that $\am Z$ is changed when $z$ describes $\gamma_{1}$.

• The Proof that every Equation has a Root

  But the latter contour evidently lies inside the circle whose centre is $a$ and whose radius is $\frac{1}{2}\rho$, and this circle does not include the origin. Hence the amplitude of $Z$ is unchanged.

• The Proof that every Equation has a Root

  We can then show, by an argument similar to that used above, that $\am(1 + \rho)$ is unchanged as $z$ describes $\gamma$ in the positive sense, while $\am z^{n}$ on the other hand is increased by $2n\pi$. Hence $\am Z$ is increased by $2n\pi$, and the proof that $Z = 0$ has a root is completed.

• The Proof that every Equation has a Root

  This assumption is obviously legitimate, for to suppose the contrary, at any stage of the argument, is to admit the truth of the theorem.

• The Proof that every Equation has a Root

  There is another proof, proceeding on different lines, which is often given. It depends, however, on an extension to functions of two or more variables of the results of [§§]102 *et seq.*

Equations

• The Proof that every Equation has a Root

  Z = P(z) = \alpha_{0} z^{n} + \alpha_{1} z^{n-1} + \dots + \alpha_{n}

  The polynomial Z = P(z) is a sum of powers of z with constant coefficients, the highest power being z^n.

• The Proof that every Equation has a Root

  |Z| = |P(x + iy)|

  The modulus of Z is the modulus of the polynomial evaluated at x + iy, and it is a positive continuous function of x and y.

• The Proof that every Equation has a Root

  P(x_{0} + iy_{0}) = 0

  The point x_0 + i y_0 is a root of the polynomial P, so P has a root in the complex plane.

• The Proof that every Equation has a Root

  |P(x + iy) - P(x_{0} + iy_{0})| < \tfrac{1}{2}\rho

  Near the point x_0 + i y_0, the value of P stays within half of rho of its value at that point, by continuity of P.

• The Proof that every Equation has a Root

  P(x + iy) = a + \phi

  Near x_0 + i y_0, P(x + iy) equals the constant a plus a small correction phi.

• The Proof that every Equation has a Root

  \delta_{m} = \delta_{1}/2^{m-1}

  One admissible choice of the decreasing sequence of side lengths delta_m is halving at each step.

• The Proof that every Equation has a Root

  z = z_{0} + \zeta

  The variable z is written as the fixed point z_0 plus a displacement zeta.

• The Proof that every Equation has a Root

  |\zeta| = \rho

  The displacement zeta is taken to have modulus rho, so z moves on a circle of radius rho about z_0.

• The Proof that every Equation has a Root

  P(z) = P(z_{0}) + A_{1}\zeta + A_{2}\zeta^{2} + \dots + A_{n}\zeta^{n}

  Expanding P in powers of zeta about z_0 gives the polynomial with coefficients A_k.

• The Proof that every Equation has a Root

  |A_{k}| = \mu

  The modulus of the first nonvanishing expansion coefficient A_k is named mu.

• The Proof that every Equation has a Root

  |A_{k+1}|\rho + |A_{k+2}|\rho^{2} + \dots + |A_{n}|\rho^{n-k} < \tfrac{1}{2}\mu

  The radius rho can be chosen small enough that the higher-order terms of the expansion total less than half of mu.

• The Proof that every Equation has a Root

  |P(z) - P(z_{0}) - A_{k}\zeta^{k}| < \tfrac{1}{2}\mu\rho^{k}

  The terms of P beyond the k-th differ from zero by less than half of mu rho^k on the small circle.

• The Proof that every Equation has a Root

  |P(z)| < |P(z_{0} + A_{k}\zeta^{k}| + \tfrac{1}{2}\mu\rho^{k}

  Bounds |P(z)| by the modulus of P(z_0) + A_k zeta^k plus half of mu rho^k. FLAG: the source has an unclosed modulus bar, |P(z_{0} + A_{k}\zeta^{k}|, which is probably a Gutenberg transcription or printing slip for |P(z_{0}) + A_{k}\zeta^{k}|; the intended form is inferred, not checked against the printed page.

• The Proof that every Equation has a Root

  |P(z_{0}) + A_{k}\zeta^{k}| = |P(z_{0})| - \mu\rho^{k}

  At k points on the circle, the modulus of P(z_0) + A_k zeta^k equals |P(z_0)| minus mu rho^k, because that circle passes through the origin-side point.

• The Proof that every Equation has a Root

  |P(z)| < |P(z_{0})| - \mu\rho^{k} + \tfrac{1}{2}\mu\rho^{k}

  Combining the estimates, |P(z)| is less than |P(z_0)| minus half of mu rho^k at some point of the circle, contradicting that m is the lower bound.

• The Proof that every Equation has a Root

  |P(z)| \to \infty

  The modulus of P(z) grows without bound as |z| grows without bound.

• The Proof that every Equation has a Root

  Z = a_{0} z^{n} \left(1 + \frac{a_{1}}{a_{0}z} + \frac{a_{2}}{a_{0} z^{2}} + \dots + \frac{a_{n}}{a_{0} z^{n}}\right)

  For large |z| the polynomial is a_0 z^n times a factor close to 1, which is the form used to find a circle on which the amplitude of Z increases by 2n pi.

• The Proof that every Equation has a Root

  Z = a_{0} z^{n} (1 + \rho)

  On the circle of radius R, Z equals a_0 z^n times (1 + rho), with rho a small correction.

• The Proof that every Equation has a Root

  \frac{|a_{1}|}{|a_{0}| R} + \frac{|a_{2}|}{|a_{0}| R^{2}} + \dots + \frac{|a_{n}|}{|a_{0}| R^{n}} < \delta

  Choosing R large enough makes the sum of scaled coefficient moduli smaller than any positive delta.

• The Proof that every Equation has a Root

  f'(z) = f(z) \left(\frac{1}{z - z_{1}} + \frac{1}{z - z_{2}} + \frac{1}{z - z_{3}}\right)

  The derivative of a cubic f equals f times the sum of reciprocals of z minus each root, used as a hint in Exercise 8.

Problems