Question 5.28: (a) Let F(z) be analytic inside and on a simple closed curve...
(a) Let F(z) be analytic inside and on a simple closed curve C except for a pole of order m at z = a inside C. Prove that
\frac{1}{2 \pi i} \oint_C F(z) d z=\lim _{z \rightarrow a} \frac{1}{(m-1) !} \frac{d^{m-1}}{d z^{m-1}}\left\{(z-a)^m F(z)\right\}(b) How would you modify the result in (a) if more than one pole were inside C?
(a) If F(z) has a pole of order m at z = a, then F(z)=f(z) /(z-a)^m where f(z) is analytic inside and on C, and f(a)≠0. Then, by Cauchy’s integral formula,
\frac{1}{2 \pi i} \oint_C F(z) d z=\frac{1}{2 \pi i} \oint_C \frac{f(z)}{(z-a)^m} d z=\frac{f^{(m-1)}(a)}{(m-1) !}=\lim _{z \rightarrow a} \frac{1}{(m-1) !} \frac{d^{m-1}}{d z^{m-1}}\left\{(z-a)^m F(z)\right\}(b) Suppose there are two poles at z = a_1 and z = a_2 inside C, of orders m_1 and m_2, respectively. Let Γ_1 and Γ_2 be circles inside C having radii ε_1 and ε_2 and centers at a_1 and a_2, respectively (see Fig. 5-12). Then
\frac{1}{2 \pi i} \oint_C F(z) d z=\frac{1}{2 \pi i} \oint_{\Gamma_1} F(z) d z+\frac{1}{2 \pi i} \oint_{\Gamma_2} F(z) d z (1)
If F(z) has a pole of order m_1 at z = a_1, then
F(z)=\frac{f_1(z)}{\left(z-a_1\right)^{m_1}} \quad \text { where } f_1(z) \text { is analytic and } f_1\left(a_1\right) \neq 0If F(z) has a pole of order m_2 at z = a_2, then
F(z)=\frac{f_2(z)}{\left(z-a_2\right)^{m_2}} \quad \text { where } f_2(z) \text { is analytic and } f_2\left(a_2\right) \neq 0Then, by (1) and part (a),
\begin{aligned}\frac{1}{2 \pi i} \oint_C F(z) d z=& \frac{1}{2 \pi i} \oint_{\Gamma_1} \frac{f_1(z)}{\left(z-a_1\right)^{m_1}} d z+\frac{1}{2 \pi i} \oint_{\Gamma_2} \frac{f_2(z)}{\left(z-a_2\right)^{m_2}} d z \\=& \lim _{z \rightarrow a_1} \frac{1}{\left(m_1-1\right) !} \frac{d^{m_1}-1}{d z^{m_1}-1}\left\{\left(z-a_1\right)^{m_1} F(z)\right\} \\&+\lim _{z \rightarrow a_2} \frac{1}{\left(m_2-1\right) !} \frac{d^{m_2}-1}{d z^{m_2}-1}\left\{\left(z-a_2\right)^{m_2} F(z)\right\}\end{aligned}If the limits on the right are denoted by R_1 and R_2, we can write
\oint_C F(z) d z=2 \pi i\left(R_1+R_2\right)where R_1 and R_2 are called the residues of F(z) at the poles z=a_1 and z=a_2.
In general, if F(z) has a number of poles inside C with residues R_1, R_2, \ldots, then \oint_C F(z) d z=2 \pi i times the sum of the residues. This result is called the residue theorem. Applications of this theorem, together with generalization to singularities other than poles, are treated in Chapter 7.
