# Unit 2: Complex Variable II (MT201) ## Power Series Representation - A **power series** $\sum a_n(z-a)^n$ has a **radius of convergence** $R = \lim\left|\dfrac{a_n}{a_{n+1}}\right|$ or $1/R = \limsup|a_n|^{1/n}$; it converges absolutely for $|z-a|R$. - Within its circle a power series is analytic and can be differentiated and integrated term by term. - Standard series: $e^z = \sum\dfrac{z^n}{n!}$, $\sin z$, $\cos z$ (all $z$); $\dfrac1{1-z} = \sum z^n$ for $|z|<1$. ## Taylor's Series - If $f$ is analytic in $|z-a|1$ write $\dfrac1{1-z} = -\sum z^{-n-1}$. - Example: $\dfrac1{z(1-z)}$ in $0<|z|<1$ equals $\dfrac1z+1+z+z^2+\cdots$. ## Singularities, Zeros and Poles - **Zero** of order $m$ at $a$: $f(z)=(z-a)^mg(z)$ with $g(a)\ne0$ ($g$ analytic). - **Isolated singularity** at $a$ classified from the Laurent series about $a$: - **Removable**: no principal part (e.g. $\sin z/z$ at 0). - **Pole of order $m$**: principal part ends at $(z-a)^{-m}$ (e.g. $1/z^2$; **simple pole** if $m=1$). - **Essential**: infinitely many negative-power terms (e.g. $e^{1/z}$ at 0). - If $f = p/q$ with $q$ having a zero of order $m$ at $a$ and $p(a)\ne0$ then $a$ is a pole of order $m$. - Singularity at infinity is studied by $z = 1/w$. ## Residues and the Residue Theorem - **Residue** of $f$ at $a$ is the coefficient $b_1$ of $(z-a)^{-1}$ in the Laurent series. - **Calculation**: - Simple pole: $\operatorname{Res}=\lim_{z\to a}(z-a)f(z)$; for $f=p/q$, $\operatorname{Res}=p(a)/q'(a)$. - Pole of order $m$: $\operatorname{Res}=\dfrac1{(m-1)!}\lim_{z\to a}\dfrac{d^{m-1}}{dz^{m-1}}\left[(z-a)^mf(z)\right]$. - **Residue theorem**: if $f$ is analytic inside and on $C$ except for finitely many singularities $a_k$ inside, $$\oint_Cf(z)\,dz = 2\pi i\sum_k\operatorname{Res}(f,a_k).$$ - Example: $\oint_{|z|=2}\dfrac{dz}{z^2+1}$ has poles $\pm i$ with residues $\tfrac1{2i}$, $-\tfrac1{2i}$ and the total is $0$. ## Real Integrals of the Type $\int_0^{2\pi}f(\cos\theta,\sin\theta)\,d\theta$ - Substitute $z=e^{i\theta}$ so $\cos\theta=\dfrac12(z+\dfrac1z)$, $\sin\theta=\dfrac1{2i}(z-\dfrac1z)$, $d\theta=\dfrac{dz}{iz}$, converting into a contour integral over the unit circle $|z|=1$. - Then $I = 2\pi i\sum\operatorname{Res}$ at poles **inside** the unit circle. - Example: $\int_0^{2\pi}\dfrac{d\theta}{a+b\cos\theta}=\dfrac{2\pi}{\sqrt{a^2-b^2}}$ for $a>|b|$. ## Bilinear Transformations - $w=\dfrac{az+b}{cz+d}$, $ad-bc\ne0$ (**Möbius** transformation). It is conformal, one-one on the extended plane, and maps circles and lines into circles or lines. - **Fixed points**: solutions of $z=\dfrac{az+b}{cz+d}$. - **Cross-ratio** $(z_1,z_2,z_3,z_4)=\dfrac{(z_1-z_2)(z_3-z_4)}{(z_2-z_3)(z_4-z_1)}$ is invariant; a transformation is determined by mapping three given points to three given points: $\dfrac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$. - Examples: $w=1/z$ (inversion), $w=e^{i\alpha}z$ (rotation), $w=z+b$ (translation). - **[Figure: Inversion]** Circle through the origin mapped to a straight line by $w=1/z$. ## Likely Exam Questions 1. Define a pole and a removable singularity. (2 marks) 2. State the Residue theorem. (2 marks) 3. Find the residue of $\dfrac{1}{z(z-1)}$ at $z=0$. (2 marks) 4. Expand $f(z)=\dfrac1{(z-1)(z-2)}$ in Laurent series for $1<|z|<2$. (long) 5. Evaluate $\int_0^{2\pi}\dfrac{d\theta}{5+4\cos\theta}$ by contour integration. (long) 6. Find the bilinear map sending $0,1,\infty$ to $-1,-i,1$. (long)