Notes on Whittaker & Watson, Chapter VII, part 2

Whittaker & Watson, A Course of Modern Analysis. Chapter VII, The Expansion of Function in Infinite Series.

7.4 When the function $f$ has poles at $a_1, a_2, \cdots$ with residues $b_1, b_2, \cdots$ located within the circles of radius $R_n\to\infty$ such that $\forall m\le n: |a_m|<R_n$ and $f$ bounded on these circles, the value of the function $f(z)$ can be expressed as the sum $$ f(z)=f(0)+\sum_{n=1}^\infty b_n\left(\frac{1}{z-a_n}+\frac{1}{a_n}\right), $$ by Cauchy’s theorem integrating on these circles.

7.5 With the result from §7.4 applied to $\frac{f'(z)}{f(z)}$, $f(z)$ can thus be expressed as the infinite sum $$ f(z)=f(0)e^{\frac{f'(0)}{f(0)}z}\prod_{n=1}^\infty\left(1-\frac{z}{a_n}\right)e^{\frac{z}{a_n}} $$ where $a_n$ are the only zeros of $f$ all being of order 1, and consequently the poles of $\frac{f'(z)}{f(z)}$ with residue $+1$.

7.6 Weierstrass factorization theorem. For a function $f(z)$ with zeros and poles at ${a_n}$ each of order $m_n$ (positive for zeros, negative for poles) in increasing distances from the $0$, $$ f(z)=f(0)e^{G(z)}\prod_{n=1}^\infty\left[\left(1-\frac{z}{a_n}\right)e^{g_n(z)}\right]^{m_n}, $$ in which $g_n=\sum\left(\frac{z}{ka_n}\right)^k$ are the polynomials such that the infinite product series converges, and $G$ expressed by the Taylor series, since $f(z)/{\prod_{n=1}^\infty\left[\left(1-\frac{z}{a_n}\right)e^{g_n(z)}\right]^{m_n}}$ is an entire function with no zeros.

7.7 For the periodic function $f(x+iy)=f(x+\pi+iy)$ which converges uniformly to $l$ if $y\to+\infty$ and to $l'$ if $y\to-\infty$, the integration $\frac{1}{2\pi i}\int f(t)\cot(t-z) dt$ on the contour of a rectangle with two sides on $x=0$ and $x=\pi$ gives $$ f(z)=\frac{l-l'}{2}+\sum_{r=1}^nc_r\cot(z-a_r), $$ where $a_r$ are poles in the strip $0<x\le\pi$ with residues $c_r$.