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 […]

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 […]

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, $1.|z|\le 3$ $2. |z|<\sqrt{3}$ $3.|z-1|<\sqrt{3}$ $4.|z-1|\le \sqrt{3}$ 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?

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 […]

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

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 […]

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 $$ \int_\mathbb{R} {\cos\left(x\right) \over 1 + x^{2}}\,{\rm e}^{−{\rm i}xt}\,{\rm d}x\,, \qquad \forall\ t > 0 $$ 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.

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?

Intereting Posts

What does “working mathematician” mean?
identity of polylogarithm
Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.
Complex polynomial and the unit circle
Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $
Measure of an elementary set in terms of cardinality
Proof that $\Bbb Z$ has no other subring than itself
Is the Galois group associated to a random polynomial solvable with probability 0?
Why are nonsquare matrices not invertible?
Is this set uncountable or countable?
What are some things we can prove they must exist, but have no idea what they are?
How to find a closed form formula for the following recurrence relation?
Given the Cauchy's problem: $y'' = 1, y(0) = 0, y'(0) = 0$. Why finite difference method doesn't agree with recurrence equation?
Examples where $H\ne \mathrm{Aut}(E/E^H)$