Articles of complex analysis

Rational function with absolute value $1$ on unit circle

What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeros and poles related to each other? So, write $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in $z$. The condition specifies that $|R(z)|=1$ for all $z$ such that $|z|=1$. In other words, $|P(z)|=|Q(z)|$ for […]

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, since it has no singularity. And what is also the correct contour for this one? Thanks in advance and more power.

Image under an entire function.

Let $f$ be an entire function and $B$ be a bounded open set in $\mathbb {C} $. Prove that boundary of image of $B$ under $f$ is contained in image of boundary of $B$. Does the same result is true for unbounded open set in $\mathbb C.$

Applications of conformal mapping

The idea is through conformal transformations satisfying the conditions requested of the problem make this an easier problem to deal,but i don’t know which be this transformation. I would appreciate any hint how to solve this. thank you very much

Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$ \int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx=\sqrt\frac{\pi}{2}\frac{\gamma \exp\big(-\alpha\sqrt{\gamma^2+\beta^2}\big)}{\sqrt{\beta^2+\gamma^2}\sqrt{\beta+\sqrt{\beta^2+\gamma^2}}}, $$ $$ \mathcal{Re}(\alpha,\beta,\gamma> 0). $$ I do not know how to approach it because of all the square root functions. It seems if $x=\pm i\alpha \ $ we may have some convergence problems because of the denominator. Perhaps there are ways […]

About the limit of the coefficient ratio for a power series over complex numbers

This is my first question in mathSE, hope that it is suitable here! I’m currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) that I have no idea where to start. Suppose $f$ is holomorphic in an open set $\Omega$ that contains the closed […]

Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}$

How to prove the following equality? $$\sum_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin’s Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that $E=\{x\in \mathbb{R}\mid f(x)=0\}$. Since $E^c$ is open in $\mathbb{R}$, $E^c$ is a countable disjoint union of open intervals $(a_i,b_i)$. On an interval $(a,b)$, if we let […]

Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$

I know that $$\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$$ For substituting $u=2$ into $$\zeta(u)\Gamma(u)=\int^\infty_0\frac{x^{u-1}}{e^x-1}dx$$ However, I suspect that there is an easier proof, maybe by the use of complex analysis. I haven’t learnt the zeta function yet. All I know is the above formula and $\zeta(2)=\frac{\pi^2}{6}$. Can we can use the above integral to find out some of the […]

Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.

I am asked to prove this theorem: If $f:U \rightarrow C$ is holomorphic in $U$ and invertible, $P\in U$ and if $D(P,r)$ is a sufficently small disc about P, then $$f^{-1}(w) = \frac{1}{2\pi i} \oint_{\partial D(P,r)}{\frac{sf'(s)}{f(s)-w}}ds$$ The book says to “imitate the proof of the argument principle” but I am not seeing the connection.