Complex Analysis

Derives Proposition 8.2. The field axioms are direct computations with Equation (8.1); associativity and distributivity reduce to real arithmetic, and \(z\,\bar z/\abs{z}^{2} = \abs{z}^{2}/\abs{z}^{2} = 1\) exhibits the inverse. Multiplicativity: \(\abs{zw}^{2} = zw\,\overline{zw} = z\bar z\, w\bar w = \abs{z}^{2}\abs{w}^{2}\), using \(\overline{zw} = \bar z\,\bar w\) (a computation). Triangle inequality: with the Euclidean identification this is the triangle inequality of \(\R^{2}\) (Equation (6.12)); algebraically, \(\abs{z+w}^{2} = \abs{z}^{2} + \abs{w}^{2} + 2\Re(z\bar w) \le \abs{z}^{2} + \abs{w}^{2} + 2\abs{z}\abs{w}\) since \(\Re u \le \abs{u}\).

Derives Proposition 8.4. Insert \(z = \ii\theta\) into Equation (8.2) and split even and odd powers, using \(\ii^{2n} = (-1)^{n}\) and \(\ii^{2n+1} = (-1)^{n}\ii\): absolute convergence permits the rearrangement (Proposition 7.47), and the two sub-series are exactly \(\cos\theta\) and \(\ii\sin\theta\). The polar form is plane trigonometry: \((x,y) = \abs{z}(\cos\theta, \sin\theta)\) for the angle \(\theta\) of the point.

Derives Theorem 8.6. Necessity. Take the limit Equation (8.4) first along real \(h = t\): \(f'(z_0) = \pp_x u + \ii\,\pp_x v\). Then along imaginary \(h = \ii t\):

Equating real and imaginary parts of the two expressions gives Equation (8.5).

Sufficiency. \(C^{1}\) regularity makes \((u,v)\) differentiable as a map \(\R^{2} \to \R^{2}\) (Definition 7.66), so

with \(h = \eta + \ii\xi\). Substituting Equation (8.5) turns the second coefficient into \(\ii\left(\pp_x u + \ii\,\pp_x v\right)\), so the two terms combine into \(\left(\pp_x u + \ii\,\pp_x v\right)(\eta + \ii\xi) = \left(\pp_x u + \ii\,\pp_x v\right)h\); dividing by \(h\) and letting \(h \to 0\) kills the \(o(\abs{h})/h\) remainder.

Derives Lemma 8.9. Write \(\int_\gamma f\,\dd z = R\,\ee^{\ii\varphi}\) with \(R \ge 0\). Then \(R = \Re\int_a^b \ee^{-\ii\varphi} f(z(t))\,z'(t)\,\dd t \le \int_a^b \abs{f(z(t))}\,\abs{z'(t)}\,\dd t \le M L\), using \(\Re u \le \abs{u}\) and monotonicity of the real integral.

Derives Proposition 8.10. \(\dv{}{t}F(z(t)) = F'(z(t))\,z'(t)\) by the chain rule, so Equation (8.6) is \(\int_a^b \dv{}{t}F(z(t))\,\dd t = F(z(b)) - F(z(a))\) by the fundamental theorem of calculus (Theorem 7.43), applied to real and imaginary parts.

Derives Lemma 8.11. Parametrize \(z = a + r\ee^{\ii\theta}\), \(\theta \in [0, 2\pi]\), \(\dd z = \ii r\ee^{\ii\theta}\dd\theta\):

For \(n \neq -1\) the integrand is the derivative of \(\ee^{\ii(n+1)\theta}/\bigl(\ii(n+1)\bigr)\), periodic over \([0,2\pi]\), so the integral vanishes; for \(n = -1\) it is \(\ii\int_0^{2\pi}\dd\theta = 2\pi\ii\).

Derives Theorem 8.12. Separate real and imaginary parts with \(f = u + \ii v\) and \(\dd z = \dd x + \ii\,\dd y\):

Green's theorem (Theorem 7.99) converts the two circulations to

and both integrands vanish identically by the Cauchy–Riemann equations Equation (8.5).

Derives Theorem 8.13. Write \(I(\Delta) = \oint_{\pp\Delta} f\,\dd z\) and let \(L\) and \(d\) be the perimeter and the diameter of \(\Delta\).

Quadrisection. Joining the midpoints of the three sides divides \(\Delta\) into four closed triangles \(\Delta^{(1)}, \dots, \Delta^{(4)}\), each similar to \(\Delta\) with ratio \(1/2\), hence of perimeter \(L/2\) and diameter \(d/2\). Traversing the four boundaries coherently, every interior edge is covered twice in opposite senses and cancels, so

and at least one of the four obeys \(\abs{I(\Delta^{(j)})} \ge \abs{I(\Delta)}/4\); call it \(\Delta_{1}\). Iterating the construction produces nested closed triangles \(\Delta = \Delta_{0} \supseteq \Delta_{1} \supseteq \cdots\) with

The limit point. Pick \(z_{n} \in \Delta_{n}\). For \(m \ge n\) both \(z_{m}\) and \(z_{n}\) lie in \(\Delta_{n}\), so \(\abs{z_{m} - z_{n}} \le d_{n} \longrightarrow 0\) and the sequence is Cauchy; by completeness of \(\C\) (Section 6.3) it converges to some \(z_{0}\), which belongs to every \(\Delta_{n}\), each \(\Delta_{n}\) being closed and containing all later terms. In particular \(z_{0} \in \Omega\).

The estimate. Complex differentiability at \(z_{0}\) (Equation (8.4)) means that

with \(\eta(z) \longrightarrow 0\) as \(z \longrightarrow z_{0}\) and \(\eta(z_{0}) = 0\). The affine part of Equation (8.11) is the derivative of the polynomial \(f(z_{0})\,z + \frac{1}{2}f'(z_{0})\,(z - z_{0})^{2}\), so its integral over the closed contour \(\pp\Delta_{n}\) vanishes (Proposition 8.10), leaving

Given \(\varepsilon > 0\), choose \(\rho > 0\) with \(\abs{\eta(z)} \le \varepsilon\) whenever \(\abs{z - z_{0}} < \rho\), and take \(n\) large enough that \(d_{n} < \rho\). Since \(z_{0} \in \Delta_{n}\), every \(z\) on \(\pp\Delta_{n}\) satisfies \(\abs{z - z_{0}} \le d_{n}\), so the integrand is bounded by \(\varepsilon\,d_{n}\) and the ML estimate (Lemma 8.9) gives

Combined with Equation (8.10) this yields \(\abs{I(\Delta)} \le \varepsilon\,d\,L\) for every \(\varepsilon > 0\), whence \(I(\Delta) = 0\).

Derives Corollary 8.15. Join the two contours by a cut; the boundary of the cut annulus is \(\gamma_1 - \gamma_2\) plus the cut traversed both ways, whose contributions cancel. Cauchy's theorem on the annulus gives \(\oint_{\gamma_1} - \oint_{\gamma_2} = 0\). In a simply connected domain every closed contour bounds, and Theorem 8.12 applies directly.

Derives Theorem 8.16. The integrand is holomorphic between \(C\) and any small circle \(C_\varepsilon : \abs{z - z_0} = \varepsilon\), so by deformation (Corollary 8.15)

The first integral is \(2\pi\ii\) by Lemma 8.11. In the second, continuity of \(f\) at \(z_0\) bounds the integrand by \(\varepsilon^{-1}\sup_{C_\varepsilon}\abs{f - f(z_0)}\), so the ML estimate (Lemma 8.9) bounds the integral by \(2\pi\sup_{C_\varepsilon}\abs{f - f(z_0)} \longrightarrow 0\) as \(\varepsilon \to 0\). Since the left side does not depend on \(\varepsilon\), it equals \(2\pi\ii f(z_0)\).

Derives Theorem 8.17. For \(n = 1\), form the difference quotient of Equation (8.12) at \(z_0\) and \(z_0 + h\):

an algebraic identity of the two kernels. As \(h \to 0\) the integrand converges uniformly on \(C\) to \(f(z)/(z-z_0)^2\) (the denominators are bounded away from zero for \(z_0\) interior and \(\abs{h}\) small), so the ML estimate carries the limit inside, giving Equation (8.13) with \(n = 1\). Induction repeats the argument on each successive formula. The estimate follows from Lemma 8.9 with \(\abs{z - z_0} = r\) and \(L = 2\pi r\).

Derives Theorem 8.18. If \(\abs{f} \le M\) everywhere, the Cauchy estimate with \(n = 1\) on a circle of radius \(r\) about any \(z_0\) gives \(\abs{f'(z_0)} \le M/r \longrightarrow 0\) as \(r \to \infty\). So \(f' \equiv 0\), and (integrating along any path, Proposition 8.10) \(f\) is constant.

Derives Theorem 8.19. If \(p\) had no root, \(1/p\) would be holomorphic on \(\C\) (quotient rule). For large \(\abs{z}\) the leading term dominates, \(\abs{p(z)} \ge \frac12\abs{c_n}\abs{z}^{n} \longrightarrow \infty\), so \(1/p \to 0\) at infinity and, being continuous on the (compact, Theorem 6.11) complement of the large-\(\abs{z}\) region, is bounded on \(\C\). Liouville (Theorem 8.18) forces \(1/p\) constant — contradicting non-constancy of \(p\). Dividing out roots inductively, \(p\) factors completely: \(\C\) is algebraically closed.

Derives Theorem 8.20. Fix \(z\) with \(\abs{z - a} = \rho < r < R\) and integrate over \(C : \abs{w - a} = r\). Expand the Cauchy kernel geometrically (Proposition 7.46), uniformly on \(C\) since \(\abs{(z-a)/(w-a)} = \rho/r < 1\):

Insert into Equation (8.12) and exchange sum and integral (uniform convergence, ML estimate); the coefficient integrals are Equation (8.13) evaluated at \(a\), giving Equation (8.14).

Derives Theorem 8.21. For \(z\) in the annulus choose radii \(r < r_1 < \abs{z - a} < r_2 < R\). The cut-annulus argument of Corollary 8.15 applied to the Cauchy formula gives

On the outer circle expand the kernel as in Theorem 8.20 (powers \((z-a)^{n}\), \(n \ge 0\)); on the inner circle the roles reverse, \(\abs{(w-a)/(z-a)} < 1\), giving \(-1/(w-z) = \sum_{m \ge 0}(w-a)^{m}/(z-a)^{m+1}\), i.e. the negative powers. Deforming both coefficient circles to the common \(C\) (Corollary 8.15) yields Equation (8.15). Uniqueness: integrating \(\sum c_n (z-a)^n\) against \((z-a)^{-m-1}\) term by term over \(C\) kills every term except \(n = m\) by Lemma 8.11, returning \(c_m\).

Derives Proposition 8.23. \(h(z) = h'(a)(z-a)\left[1 + O(z-a)\right]\) by its Taylor expansion (Theorem 8.20), so \(f(z) = \dfrac{g(a)}{h'(a)}\,(z-a)^{-1} + (\text{holomorphic part})\); the coefficient of \((z-a)^{-1}\) is read off.

Derives Theorem 8.24. Excise from \(D\) a small disc \(\abs{z - a_j} \le \varepsilon\) around each singularity. On the excised region \(f\) is holomorphic, so Cauchy's theorem in the cut-boundary form (Corollary 8.15) gives

with \(C_{\varepsilon,j}\) the small circles, counterclockwise. On each, insert the Laurent series of \(f\) about \(a_j\) (Theorem 8.21), which converges uniformly on \(C_{\varepsilon,j}\), and integrate term by term: by Lemma 8.11 every power dies except \((z - a_j)^{-1}\), which contributes \(2\pi\ii\,\operatorname{Res}_{z=a_j} f\).