# Unit 8: Series and Residues (MT324) ## Convergence of Sequences and Series - A sequence $z_n=x_n+iy_n$ converges to $z=x+iy$ iff $x_n\to x$ and $y_n\to y$. - A series $\sum z_n$ converges if the partial sums $S_N$ converge; necessary condition $z_n\to0$. - **Absolute convergence:** $\sum|z_n|$ converges; this implies convergence. - **Tests:** comparison, ratio, root tests. - **Geometric series:** $\sum_{n=0}^\infty z^n=\dfrac1{1-z}$ for $|z|<1$. ## Taylor Series **Theorem:** if $f$ is analytic in the disc $|z-z_0|1$. - Example: $e^{1/z}=\sum_{n\ge0}\dfrac{1}{n!z^n}$. ## Absolute and Uniform Convergence of Power Series - Each power series $\sum a_n(z-z_0)^n$ has a **radius of convergence** $R=\dfrac1{\limsup|a_n|^{1/n}}$. - Converges **absolutely** for $|z-z_0|R$. - Converges **uniformly** on every closed disc $|z-z_0|\le rm$; $\lim|f|=\infty$; e.g. $\dfrac1{z^2}$; **simple pole** $m=1$. - **Essential:** infinitely many negative-power terms; e.g. $e^{1/z}$ at $0$ (Picard: takes nearly every value). - A zero of order $m$ of $g$ gives a pole of order $m$ of $1/g$. ## Residues The **residue** of $f$ at an isolated singularity $z_0$ is the coefficient $a_{-1}$ of $(z-z_0)^{-1}$ in the Laurent series: $\operatorname{Res}_{z_0}f=\dfrac1{2\pi i}\oint_Cf\,dz$. ## Residues at Poles - **Simple pole:** $\operatorname{Res}=\lim_{z\to z_0}(z-z_0)f(z)$. If $f=\dfrac{p}{q}$ with $q(z_0)=0$, $q'(z_0)\ne0$: $\operatorname{Res}=\dfrac{p(z_0)}{q'(z_0)}$. - **Pole of order $m$:** $$\operatorname{Res}_{z_0}f=\frac1{(m-1)!}\lim_{z\to z_0}\frac{d^{m-1}}{dz^{m-1}}\big[(z-z_0)^mf(z)\big].$$ - Examples: $\operatorname{Res}_{z=1}\dfrac{e^z}{z-1}=e$; $\operatorname{Res}_{z=i}\dfrac1{z^2+1}=\dfrac1{2i}$; $\operatorname{Res}_{z=0}\dfrac{e^z}{z^2}=1$. ## Cauchy's Residue Theorem If $f$ is analytic inside and on a simple closed positively oriented contour $C$, except for finitely many singular points $z_1,\dots,z_n$ inside $C$: $$\oint_Cf(z)\,dz=2\pi i\sum_{k=1}^n\operatorname{Res}_{z_k}f.$$ - Example: $\oint_{|z|=2}\dfrac{dz}{z^2+1}=2\pi i\Big(\dfrac1{2i}-\dfrac1{2i}\Big)=0$. - Use: evaluating real integrals such as $\int_0^{2\pi}\dfrac{d\theta}{a+b\cos\theta}$. ## Residue at Infinity $$\operatorname{Res}_{\infty}f=-\frac1{2\pi i}\oint_{C}f(z)\,dz=-\operatorname{Res}_{w=0}\Big[\frac1{w^2}f\Big(\frac1w\Big)\Big]$$ (with $C$ large, positively oriented). - Sum of all finite residues $+$ residue at infinity $=0$. - Example: for $f=1/z$, $\operatorname{Res}_\infty=-1$. | Singularity | Laurent principal part | Behaviour of $f$ | |---|---|---| | Removable | none | finite limit | | Pole order $m$ | $m$ terms | $\to\infty$ | | Essential | infinite terms | no limit | **[Figure: Annulus of convergence of a Laurent series]** Two concentric circles about $z_0$ with the annulus shaded, and a contour $C$ inside it. ## Likely Exam Questions 1. Define a removable singularity and a pole. (2 marks) 2. State Cauchy's residue theorem. (2 marks) 3. Expand $f(z)=\dfrac1{z(z-1)}$ in Laurent series in $0<|z|<1$. 4. Find the Taylor series of $\sin z$ and $\dfrac1{1-z}$ about $0$. 5. Find the residue of $\dfrac{z^2}{(z-1)^2(z+2)}$ at its poles. 6. Evaluate $\oint_{|z|=2}\dfrac{e^z}{z(z-1)}dz$ using the residue theorem.