The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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0
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1answer
46 views

Entire function with $f(0)=1$ and $f(1-1/n^2)=0$

Can we find an entire function with the properties $f(0)=1$ and $f(\frac{n^2-1)}{n^2})=0$ We could have applied the Weirestrass Theorem to find such entire function with prescribed zeros.But , the ...
1
vote
0answers
28 views

Determining a meromorphic function from its poles

I am working on an old qual question. Suppose we have a function $f(z)$ meromorphic on the extended complex plane. The only poles are $-1$, which has principal part $1/(z+1)$, and $2$, which has ...
1
vote
1answer
29 views

finding the inverse Laplace transform of $\frac{1}{z\sqrt{z+1}}$

i know that the inverse Laplace transform is given by $$2\pi i \left\{\sum\space\text{ of the residues at the poles of}\space e^{zt}f(z)\right\}- \frac{1}{2 \pi i}\int \text{ along the branch cut}$$ ...
11
votes
6answers
757 views

Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
0
votes
1answer
45 views

What is the radius of convergence of the power series $\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$?

What is the radius of convergence of the power series? $$\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$$ Progress Used the ratio test, but got $0$ from it.
2
votes
0answers
27 views

about the ratio of the coefficients [duplicate]

Let $f$ be holomorphic on an open disk containing the unit circle, except in a pole $w$ on the unit circle. Let $\displaystyle \sum_{k=0}^{\infty} a_n z^n$ be its expansion. Show that ...
2
votes
1answer
180 views

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum's ...
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1answer
43 views

how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ [on hold]

calculate the following integral $\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ I need to very hollowing steps.thank you in advance
283
votes
7answers
8k views
+400

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
2
votes
0answers
22 views

Dirichlet problem of complex analysis

$\Omega=\{z\in \mathbb{C}:|z|<2, |z-1|>1\}$ I have to find a harmonic function $\phi$ such that $\phi=1$ on the outer boundary and $\phi=3$ in the inner boundary and $(z\neq 2)$ How do I map ...
2
votes
0answers
25 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
0
votes
1answer
29 views

plotting $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under mapping $w=\sin(z)$

i need to plot this $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under $w=\sin(z)$ mapping so what i did is $ y=0 , \frac{-\pi}{2}<x<\frac{\pi}{2} => -1<u<1 , v=0 $ $ y=1 ...
5
votes
1answer
116 views
+100

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
1
vote
1answer
38 views

Two continuous functions such that $f(z)^2=g(z)^2=1-z^2$

I need to find a domain $D$ and two continuous functions $f$ and $g$ such that $$f(z)^2=g(z)^2=1-z^2$$ for all $z\in D$. This is what I tried. Consider $$h(z)=e^{{1/2}L(z)}$$ where $L$ is a branch ...
1
vote
2answers
54 views

Is it true that a continuous function with compact support is uniformly continuous?

I've been trying to prove the given $f:\mathbb R\rightarrow \mathbb C$ continuous with compact support, $f$ is uniformly continuous. I don't know if it's true or not, but it is highly plausible and ...
4
votes
4answers
145 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
4
votes
3answers
186 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis? ...
4
votes
3answers
164 views

Find solution of equation $(z+1)^5=z^5$ [duplicate]

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
0
votes
1answer
24 views

Using $e^{ix}$ instead of sine and cosine in contour integration

A while ago I asked: Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis. Instead of using $\cos(z)$ an answerer said that is valid to use $e^{ix}$ How is this valid? I dont ...
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votes
0answers
38 views

Vanishing fourier coefficients

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. How can I show that $$\widehat{g}(l)=0 \ \text{for} \ l \not\equiv 0 \ \text{mod} \ n?$$ $\widehat{g}(l)$ is defined ...
5
votes
3answers
71 views

Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis.

Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Using only complex analysis. $$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 ...
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vote
0answers
20 views

Not Univalent on the open disk, but univalent on any annulus [duplicate]

This is a problem from old qualifying exam. Let $f$ be an analytic function on the open unit disk $D$. Suppose that $f$ is not univalent on $D$. Show that $f$ cannot be univalent on any ...
2
votes
0answers
23 views

Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
6
votes
2answers
97 views

Evaluating sums using residues $(-1)^n/n^2$ [duplicate]

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
3
votes
0answers
16 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
0
votes
0answers
13 views

Harmonic function condition for $ v=f(x,y)$ [on hold]

I know that to see if $u=F(x,y$) is a harmonic function $Uxx+Uyy=0$ but if instead of U function I have the V function $v=F(x,y)$ is it still $Vxx+Vyy=0$ or I should check something else , ...
0
votes
0answers
6 views

the bilinear mapping that maps the given three points onto the three given points in the respective order 1,i,-5 onto i,-2i,2

i need to solve this question : the bilinear mapping that maps the given three points onto the three given points in the respective order 1,i,-5 onto i,-2i,2 , the way i can think of is mobius ...
0
votes
0answers
21 views

Branch of the cube root function in $\mathbb{C} \backslash [0,\infty)$

Let $D=$$\mathbb{C} \backslash[0,\infty)$ and define $f:D\to \mathbb{C}$ by $$f(z) = \begin{cases} z^{1/3} & \text{if }\operatorname{Im} z \geq 0, \\[6pt] \tfrac{1}{2}(-1+i\sqrt 3)z^{1/3} & ...
3
votes
5answers
97 views

Complex Analysis book including integration

FOR BEGINNERS: Currently, I am looking for a textbook on complex analysis, which covers complex analysis from the beginning, and majorly focuses on contour integration, and the residue theorem. On ...
1
vote
2answers
96 views

Variance of imaginary part of a complex random variable.

$\def\im{\mathrm {im}}$ Let $\xi$ be a complex random variable. $\xi_\im$ is imaginary part of $\xi$. Obviously $E(\xi_{\im}) = \operatorname{Im}E(\xi)$. My question is how to compute ...
3
votes
2answers
40 views

Help identifying the singularities of $\csc(\cos z) = \frac{1}{\sin(\cos z)}$

I am really stuck with this one: $\frac{1}{\sin(\cos z)}$ has a singularity when $\cos z = k\pi $ since $\sin(k\pi) = 0$ but how do I solve for the value of $z$, how can i evaluate $\cos z = k\pi $?
1
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1answer
19 views

Derivation of energy integral - harmonic functions

I am following the solution of the following problem on the topic of the energy integral of a surface. For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the ...
0
votes
1answer
31 views

Question about the biholomorphic mapping $w = z+\frac{a^2}{z}$

I wanted to do the "show this" in this example: But I'm a bit stuck. Is there a fast way to do this? I tried by writing: $$ w = x+iy = z+\frac{a^2}{z}$$ And solving for $z$, which after some ...
3
votes
2answers
92 views

difficult complex integral $\int_\gamma \frac{1}{z^2+i}dz$

We are asked to calculate $\int_\gamma \frac{1}{z^2+i}dz$ where $\gamma$ is the straight line from $i$ to $-i$ in that direction. My parametrization is simple, I chose $z(t)=i-2it$. Notice that ...
0
votes
2answers
34 views

Absolute convergence of an infinite series and p-series test

Why does the infinite series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}z^n$ where $z\in \mathbb{C}$ converge absolutely for $|z|<1$. Doesn't the series diverge because if we apply the absolute ...
1
vote
2answers
134 views

Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle. (See http://en.wikipedia.org/wiki/Variation_of_argument) But I do not have books about it and I cannot find anything of value ...
1
vote
1answer
31 views

A function differentiable only at $0$ and for $|z|=1$

I need to find a polynomial function that is differentiable at the origin where $f'(0)=1$ and at every point $|z|=1$ but at no other point in the complex plane. I just have no clue how to solve ...
0
votes
1answer
516 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
1
vote
1answer
19 views

Complex integration Cauchy Formula

$\oint_{\left | z \right |=0.5} \frac{dz}{(z-1)(\sin z)} $ Define: $f(z) = \frac{z}{(\sin z)(z-1)}$ Define: $g(z) = \frac{f(z)}{z}$ Now integrate using Cauchy Integration Formula $\oint_{\left | ...
0
votes
0answers
14 views

Evaluate the given integral along the given (positively oriented) circle. [on hold]

Ok, so I have the following problems that I am working on. It says to evaluate 1) where C is given by |z+1|=1/2 2) where C is given by |z-2|=1/2 3) where C is given by |z|=2 4) where C ...
0
votes
0answers
61 views

If $|f|+|g|$ is constant then each of $f, g$ is constant [duplicate]

Let $f,g: U \rightarrow \mathbb{C}$ be holomorphic on the open and connected subset $U$. If $|f| + |g|$ is constant on $U$ show that $f, g$ are constant on $U$. What can we say about finite or ...
0
votes
0answers
28 views

Is $z/\sin z$ analytic in the complex plane? [on hold]

Verify if the function $$f(z) = \frac{z}{\sin z}$$ is analytic in the complex plane?
-2
votes
1answer
41 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
6
votes
1answer
252 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
0
votes
1answer
28 views

Having trouble combining Weierstrass approximation theorem and the infinite sequence of holomorphic functions

The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials. Trying to facilitate the digestion of the fatty Christmas food, I ...
0
votes
4answers
57 views

If a continuous function satisfies $|f(z)^2-1|<1$ for every $z$, then either $|f(z)-1|<1$ of $|f(z)+1|<1$ for every $z$

Suppose a continuous function $f:D\rightarrow \mathbb{C}$ where $D$ is a plane domain, has the property $|f(z)^2-1|<1$ for every $z$ in $D$. Show that $|f(z)-1|<1$ of $|f(z)+1|<1$ for every ...
0
votes
0answers
41 views

Complex analysis (Analytic function, sharp upper bound)

I encouter complex analysis problems the I think it is quite to do. Could anyone please give a hint or guideline. Thank you very much in advanced. Let $D$ be an open unit disc $\{z \in \mathbb{C}| ...
0
votes
2answers
97 views

Contour integral of $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$

I'm supposed to evaluate $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$ Using a contour of a unit circle, $z=e^{i\theta}$. This is the same as: $$2i \oint \frac{1}{z^2 - Az + 1 } dz $$ The ...
5
votes
0answers
114 views

How can $ i $ be distinguished from $ - i $? [duplicate]

Mathematicians designate one solution to $x^2 = -1$ as $i$ and the other as $-i$. Would anybody notice if we switched their identities? Any polynomial $p(x)$ with a complex root will also have its ...
0
votes
2answers
64 views

Help Finding the Cauchy Principle Value of $\int_{0}^{2\pi}\frac{d\theta}{1+2cos(\theta)}$

$$\int_{0}^{2\pi}\frac{d\theta}{1+2cos(\theta)}$$ My attempt: parametrise using $z=e^{i\theta}$ (i think we always use a unit circle for CPV's) $\therefore dz = ie^{i\theta}d\theta$ $\implies ...