Articles of complex analysis

Convergence power series in boundary

Say I have a power series $\sum_{k=0}^\infty a_k z^k$ with radius of convergence $0<R<\infty$. What can be said topologically about the set $\{z\in\Bbb C\mid |z|=R\,\mbox{ and }\sum_{k=0}^\infty a_k z^k \mbox{ converges}\}$ ? Is it possible for there to be an isolated point? Is it possible that it converges for only one point in the boundary […]

Location of zeros of a sum of exponentials

Describe the approximate locations of the zeros of the function $$ f(z) = e^{iz}+e^{-iz}+e^z $$ lying outside the circle $|z|=R >>1$. Another prelim problem. For Rouche’s theorem we need to find a bounded domain. I tried for $R<|z|<r$ and then looked in the left half plane so that $|e^z|<1$ but then I cannot apply the […]

Find $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.

Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$. My Try: So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\pi\}$. On $\Gamma$, $z=a\cos t+i b \sin t$, so $|z|^2=z\bar{z}=a^2\cos^2 t+b^2 \sin^2 t$. Now, $dz=-a\sin t+i b \cos t dt$. Hence, the integral […]

Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$

How to prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?

Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am looking if I can evaluate it using Gauss MVT. ADDED:: Here is what I have got so far!! $$\ln 2 = […]

The line integral $\int_{\gamma}\frac 1z$ and branchs of logarithm

Fix $w=re^{i\theta}\neq 0$ and let $\gamma$ be a rectifiable path in $\mathbb{C}\setminus\{0\}$ from $1$ to $w$. Show that there is a $k\in\mathbb{Z}$ such that $\displaystyle\int_{\gamma}\dfrac 1z=\log r+i\theta+2\pi i k$. If we could find some primitive $F$ of $\dfrac 1z$, we would have $\displaystyle\int_{\gamma}\dfrac 1z=F(w)-F(1)$. If we could take $F(z)=\log z$, the principal branch, then $F(w)=\log […]

Proof that a certain entire function is a polynomial

Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this statement, and I think one can prove it as follows: consider $f(1/z)$. $0$ cannot be an essential singularity of $f(1/z)$, for the big Picard theorem would […]

Computing Residues

How might one go about computing the residue of $\frac {z^2 + 3z – 1}{z+2}$? I understand it has a pole at -2 and that we should then expand the numerator in powers of 2, but the book seems to do it by inspection. How does it look when done methodically? EDIT: I should clarify […]

Show that $f(z)=0$.

This question already has an answer here: on the boundary of analytic functions 2 answers