Articles of complex analysis

How to finish proof of $ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I’m trying to prove the identity $$ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I’ve done so far: From geometric series $\sum_{k=0}^{n-1}q = {1-q^n \over 1 – q}$ for $q = e^{i \varphi}$ and taking the real part on both sides I got $$ \sum_{k=0}^{n-1}\cos (k\varphi […]

Images of lines $y = k = \mbox{constant}$ under the mapping $w = \cos (z)$

I want to solve this question: find the images of lines $y = k = \mbox{constant}$ under the mapping $w =\cos(z).$ I know that $w=\cos(z)=\cos(x)\cosh(y)-i\sin(x)\sinh(y)$ so $u=\cos(x)\cosh(y)$ and $v=-\sin(x)\sinh(y)$ but $y=k$ is only a line and how can i map it? I don’t know what values should be given to x. I put k in […]

$P(z) $ is a polynomial of degree $n-1$ and $P(ω_i) = f(ω_i)$.

Let $C$ be a regular curve enclosing the distinct points $ω_1,ω_2,…ω_n$ and let $p(ω) = (ω −ω_1)(ω −ω_2) \cdots (ω −ω_n)$. Suppose that $f (ω)$ is analytic in a region that includes $C$. Show that $$P(z) = \frac{1}{2 \pi i} \int_C \frac{f(ω)}{p(ω)} . \frac{p(ω) -p(z)}{ω -z} dω$$ is a polynomial of degree $n-1$ and $P(ω_i) […]

$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when

$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when, $1.|z|\le 3$ $2. |z|<\sqrt{3}$ $3.|z-1|<\sqrt{3}$ $4.|z-1|\le \sqrt{3}$ The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number $$C=\lim\sup|3^{-n}(z-1)^{2n}|^{1\over n}$$ for convergence $C<1$ and I get $|z-1|<\sqrt{3}$, Is it okay?

Number of Solutions to $e^{z}-3z-1=0$ in the Unit Disk

I am working through some of the past qualifying exams in complex analysis and I am a bit stuck on the question I posed in the title. My immediately thought is use Rouche’s Theorem. For instance, I tried letting $f(z)=e^{z}$ and $g(z)=3z+1$ in hopes of getting $|f(z)|\leq |g(z)|$ on $|z|=1$. But this is false since […]

Contour integration of $\frac{\log( x)}{x^2+a^2}$

This question already has an answer here: Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration 5 answers

Breaking a contour integral into 3 separate contours?

We can try to integrate the following function around a counter-clockwise circular contour: $$\frac{x^3}{(x-1)(x-2)(x-3)}$$ Can someone show how to use the Cauchy–Goursat theorem (explained here and here) to break this apart into 3 separate contours? In other words, I’d like to take $$\int_c{\frac{x^3}{(x-1)(x-2)(x-3)}}$$ and get $$\int_{C_1}{f_1(x)} + \int_{C_2}{f_2(x)} + \int_{C_3}{f_3(x)}$$ I’m hoping for a pretty […]

Bound on second derivative for a function holomorphic on the disk $D(0,1)$

I am working on a problem from an old complex analysis qual, and have run across the following problem: Let $f(z)$ be holomorphic on $D(0,1)$, such that $|f(z)|\leq 1$ for all $|z|<1$. If $f(0) = f'(0) = 0$, prove $|f”(0)|\leq 2$. After working through it for awhile, I realized we definitely needs Schwarz’ Lemma for […]

Use contour integration methods to compute $\int_\mathbb{R}\frac{\cos x}{1 + x^2}e^{−ixt}dx$ for all $t > 0.$

Use contour integration methods to compute $$ \int_\mathbb{R} {\cos\left(x\right) \over 1 + x^{2}}\,{\rm e}^{−{\rm i}xt}\,{\rm d}x\,, \qquad \forall\ t > 0 $$ Could someone suggest the proper contour to use ?. I can do the rest I’m sure, but I have no clue what contour to use. Thanks.

Taylor series of an entire function which is not a polynomial

I have an entire function which is not a polynomial. Is there a way to use the Casorati-Weierstrass theorem to prove there exists a point $z_0$ such that every coefficient of the Taylor series at $z_0$ is not zero?