Articles of complex analysis

To show sum of residues of $f(z)$ over all poles is $0$

Let $p(z)$ and $q(z)$ be relatively prime polynomials with complex co-efficients so that $deg(q(z))\ge deg(p(z))+2$ and let $f(z)=p(z)/q(z)$. We need to show that the sum of residues of $f(z)$ over all poles is $0$ Well, I tried like this: by Residue theorem: If $f$ is analytic in a domain except for isolated singularities at $a_1,\dots […]

Convergence of a recursive sequence $z_{n+1} = \frac{1}{2} ( z_n – \frac{1}{z_n}) $

Let $p$ be a complex number. Let $ z_0 = p $ and, for $ n \geq 1 $, define $z_{n+1} = \frac{1}{2} ( z_n – \frac{1}{z_n}) $ if $z_n \neq 0 $. Prove the following: i) If $ \{ z_n \} $ converges to a limit $a$, then $a^2 + 1 = 0 $ […]

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto \frac{1}{\overline{z}}, $$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$.

Evaluating $ \int_{|z|=1} \frac{e^z}{(z+3)\sin(2z)} \ dz$

Consider the following integral: $\displaystyle \int_{|z|=1} \frac{e^z}{(z+3)\sin(2z)}dz$. To apply the Cauchy integral formula, I rewrite it as: $\displaystyle \int_{|z|=1} \frac{ze^z}{z(z+3)\sin(2z)}dz$ and take $\displaystyle f(z)=\frac{ze^z}{(z+3)\sin(2z)}$. The problem now is that I would compute $f(0)$ in the next step, but $\sin(2\cdot 0)=0$, so the denominator of $f$ is undefined. How would I deal with this?

Finding value of exponential sum

I’d like to find the value of the following sum $$S(u) = \sum_{n=0}^\infty \frac{e^{iu2^n}}{2^{n+1}}$$ for $u \in \mathbb R$, but I can’t seem to do it. Unfruitful work Writing $$S = \sum_{n=0}^\infty \sum_{p=0}^\infty \frac{(iu2^n)^p}{2^{n+1}p!}$$ is proven useless because I can’t exchange the order of summation. Writing $$S = \sum_{n=0}^\infty \frac{\cos(u2^n)}{2^{n+1}} + i \sum_{n=0}^\infty \frac{\sin(u2^n)}{2^{n+1}}$$ seems […]

Easy way to compute Jacobian for $f(z) = z^n$?

As a function on $\mathbb R^2$, I want to compute the Jacobian of $f(z)=z^n$. Is there an easy way to this? Write $z=x+iy$ .. and compute real part and imaginary part of $f$ and differentiate with respect to $x,y$ seems to be very tedious work…

Conformal map from unit disk to strip

I have the following question: Write down the solution $u(x, y)$ to the Dirichlet problem for the following region and boundary conditions: $U = \{x + iy : 0\le y\le1\}; u(x, 0) = 0, u(x, 1) = 1$. Hence use appropriate conformal maps to find to a solution in the following region and with the […]

Prove if $f'(1)$ is Real then $f'(1)\ge 1$

While Solving exams of previous years I encountered this problem which I cannot solve Prove if $f'(1)$ is Real then $f'(1)\ge 1$, Let $f$ be holomorphic(has a derivative) at $\Omega = \{|z|<1\}\cup\{|z-1|<r\}$ for some $r>0$. Assume: $f(0)=0$ $f(1)=1$ $\forall z\in\Omega \space(|z|<1 \implies |f(z)|<1)$ Anyway I try to look at it… I don’t get anywhere :\ […]

Non-constructive proof that $\sum_{j=1}^n j^k$ is a polynomial $p(n)$ of degree $k+1$

So it can be shown that there are special polynomials (I forget their name) $p_k$ of degree $k$ that satisfy $\sum_{j=1}^n p_k(j) = n^{k+1}$, and that these polynomials are linearly independent so that a sum of any polynomial from $j=1$ to $n$ is equal to a polynomial in $n$ of one degree higher. However I […]

Complex substitution allowed but changes result

It is well known that $$ I := \int_L \frac{1}{z} ~\text{d}z = 2 \pi i $$ where $L$ is the complex unit circle, parametrized by $\gamma(t) = e^{it}, 0 \leq t \leq 2 \pi$. However, using complex substitution, I obtain the following: by definition op complex line integrals, we have $$ I = \int_{t = […]