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, ...
131
votes
17answers
8k views
Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$
As I have heard people did not trust Euler when he first discovered the formula
$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler and he gave other proofs.
I ...
36
votes
14answers
5k views
Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$
Well, can anyone ...
20
votes
9answers
7k views
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?
thanks.
34
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
20
votes
1answer
371 views
Which sets are removable for holomorphic functions?
[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.]
Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...
3
votes
2answers
340 views
definite integral calculation with 0 pole and
$$\int_0^\infty \frac{\sin(2\pi x)}{x(x^{2}+3)}$$
I look at $\frac{e^{2\pi i z}}{z^{3}+3z}$ , also calculate the residues, but they don't get me the right answer. ( I use that for this case, it holds ...
44
votes
3answers
3k views
$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
11
votes
2answers
494 views
Infinite products - reference needed!
I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
11
votes
2answers
702 views
if $f$ is entire and $|f(z)| \leq 1+|z|^{1/2}$, why must $f$ be constant?
How can we prove that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic (analytic) and $|f(z)| \leq 1+|z|^{1/2} \forall z$, then $f$ is constant?
Liouville's theorem springs to mind, but I can't ...
28
votes
19answers
2k views
Interesting results easily achieved using complex numbers
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals ...
2
votes
1answer
286 views
Prove analyticity by Morera's theorem
Let $f$ be continuous on the complex plane and analytic on the complement of the coordinate axes. Show that $f$ is analytic everywhere. Hint: Morera's theorem. I think that I need to show that the ...
2
votes
3answers
120 views
Difficulties performing Laurent Series expansions to determine Residues
The following problems are from Brown and Churchill's Complex Variables, 8ed.
From ยง71 concerning Residues and Poles, problem #1d:
Determine the residue at $z = 0$ of the function ...
12
votes
5answers
2k views
entire 1-1 function
Can we prove that given an entire function $f$ that is also one to one then $f$ must be linear?
Thanks for any help.
4
votes
3answers
215 views
showing a function defined from an integral is entire
Let $f$ be a continuous complex-valued function on the unit interval. For any complex number $z$, define $F(z)=\int _0 ^1 f(t) e^{zt} dt$.
How do I show that $F$ is entire?
1
vote
2answers
186 views
analytic functions defined on $A\cup D$
Let $f$, $g$ be analytic function defined on $A\cup D$ where
$A = \{z \in \mathbb{C}: \frac{1}{2}<|z|<1\}$ and
$D = \{z \in \mathbb{C}: |z-2|<1\}$
Which of the following statements are true?
...
4
votes
2answers
511 views
Find laurent expansion of $\frac{z-1}{(z-2)(z-3)}$ in annulus {$z:2<|z|<3$}.
Find the Laurent expansion of $\frac{z-1}{(z-2)(z-3)}$ in annulus {$z:2<|z|<3$}.
So far I have the following, i'm not 100% sure if it is right.
$\frac{z-1}{(z-2)(z-3)}$ = ...
2
votes
5answers
404 views
Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$
How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers?
$$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$
EDIT:: ...
1
vote
2answers
163 views
Determine and classify all singular points
Determine and find residues for all singular points $z\in \mathbb{C}$ for
(i) $\frac{1}{z\sin(2z)}$
(ii) $\frac{1}{1-e^{-z}}$
Note: I have worked out (i), but (ii) seems still not easy.
25
votes
21answers
5k views
What is a good complex analysis textbook?
I'm out of college, and trying to learn complex analysis on my own. I took out Ahlfors' text from the library, but I'm finding it difficult. Any textbook recommendations? I'm probably at an ...
13
votes
5answers
803 views
Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general:
$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$
...
15
votes
1answer
524 views
If $f,g$ are both analytic and $f(z) = g(z)$ for uncountably many $z$, is it true that $f = g$?
If two analytical functions of $\mathbb{C}$ f and g are equal on an infinite number of input values, than they are equal. I can't seem to find a counterexample, but I haven't seen this anywhere ...
6
votes
1answer
171 views
Proof that a certain entire function is a polynomial
Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
32
votes
4answers
1k views
How is $\mathbb{C}$ different than $\mathbb{R}^2$?
I'm taking a course in Complex Analysis, and the teacher mentioned that if we do not restrict our attention to analytic functions, we would just be looking at functions from $\mathbb{R}^2$ to ...
14
votes
5answers
497 views
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
17
votes
6answers
1k views
Characterizing nonconstant entire functions with modulus 1 on the unit circle
Is there a characterization of the nonconstant entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?
Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is ...
6
votes
1answer
246 views
When can we find holomorphic bijections between annuli?
I'm self-studying some complex analysis, and apparently holomorphic bijections between two annuli exist precisely when the ratios of the radii are the same. More exactly, if ...
14
votes
1answer
429 views
Examples of Taylor series with interesting convergence along the boundary of convergence?
In most standard examples of power series, the question of convergence along the boundary of convergence has one of several "simple" answers. (I am considering power series of a complex variable.)
...
4
votes
1answer
284 views
If $|f(z)|\lt a|q(z)|$ for some $a\gt 0$, then $f=bq$ for some $b\in \mathbb C$
If $q\colon\mathbb{C}\to\mathbb{C}$ is a polynomial, $f\colon\mathbb{C}\to\mathbb{C}$ is analytic on all of $\mathbb{C}$, and if there exists $a\gt 0$ such that $|f(z)| \lt a|q(z)|$ for every $z\in ...
8
votes
3answers
278 views
$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant
Let $f\colon\mathbb C \to \mathbb C$ be entire. Show that if
$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ for all $z \in \mathbb C$, then $f$ is constant on $\mathbb C$. How I can answer this ...
8
votes
2answers
403 views
can $ \int_0^{\pi/2} \ln ( \sin(x)) \; dx$ be evaluated with complex integral
Can the following integral be evaluated using complex method by substituting $\sin(x) = {e^{ix}-e^{-ix} \over 2i}$?
$$ I=\int_0^{\pi/2} \ln ( \sin(x)) \; dx = - {\pi \ln(2) \over 2}$$
4
votes
5answers
527 views
How to combine complex powers?
Regarding this thread,
it is not possible to combine complex powers in the usual way:
$$ (x^y)^z = x^{yz} $$
There was mention of multi-valued functions, is there some theory that makes this all ...
5
votes
2answers
211 views
For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
For a polynomial $p(z)=a_0+\ldots+a_nz^n$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$.
I get $|p(z)|\geq |a_n|R^n$, which makes that factor of $1/2$ useless. So I ...
4
votes
1answer
366 views
Locally bounded Family
I'm studying for an exam an I came across a problem that I am having difficultly solving.
Let $\mathcal{F}$ is a family of analytic functions on the closed unit disc, $D$.
Suppose
$\int_{D} |f|^{2} ...
8
votes
2answers
253 views
Holomorphic functions and limits of a sequence
Let $f$ and $g$ be two holomorphic functions in a connected open set $D$ of the plane, which have no zeros in $D$; if there is a sequence $(a_n)$ of points of $D$ such that
$$\lim a_n = a, \qquad a ...
6
votes
1answer
263 views
Uniform convergence of infinite series
Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
2
votes
3answers
825 views
Evaluate the integral $\int_0^{2 \pi} {\cos^2 \theta \over a + b \cos \theta}\; d\theta$
Given $a > b > 0$ what is the fastest possible way to evaluate the following integral using Residue theorem. I'm confused weather to take the imaginary part of $z^2$ or whole integral.
...
1
vote
1answer
271 views
Find all Laurent series of the form…
Find all Laurent series of the form $\sum_{-\infty} ^{\infty} a_n $ for the function
$f(z)= \frac{z^2}{(1-z)^2(1+z)}$
There are a lot of problems similar to this. What are all the forms? I need to ...
46
votes
12answers
3k views
Intuitive explanation of Cauchy's Integral Formula in Complex Analysis
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
104
votes
3answers
5k views
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
56
votes
5answers
3k views
What is the Riemann-Zeta function?
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
5
votes
4answers
322 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
10
votes
2answers
450 views
Entire functions such that $f(z^{2})=f(z)^{2}$
I'm having trouble solving this one. Could you help me?
Characterize the entire functions such that $f(z^{2})=f(z)^{2}$ for all $z\in \mathbb{C}$.
Hint: Divide in the cases $f(0)=1$ and $f(0)=0$. ...
7
votes
2answers
327 views
Is there an elementary method for evaluating $\int_0^\infty \frac{dx}{x^s (x+1)}$?
I found a way to evaluate $\int_0^\infty \frac{dx}{x^s (x+1)}$ using the assumption that $s\in\mathbb{R}$ and $0<s<1$.
Apparently it should be easily extended to all $s\in\mathbb{C}$ with ...
4
votes
3answers
904 views
Riemann Zeta Function and Analytic Continuation
The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. ...
11
votes
1answer
404 views
$\int_{0}^{\infty}\frac{dx}{1+x^n}$
My goal is to evaluate $$\int_{0}^{\infty}\frac{dx}{1+x^n}\;\;(n\in\mathbb{N},n\geq2).$$
Here is my approach:
Clearly, the integral converges.
Denote the value of the integral by $I_n$.
Now let ...
8
votes
3answers
794 views
When is a function satisfying the Cauchy-Riemann equations holomorphic?
It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. ...
7
votes
2answers
560 views
Approximation of Products of Truncated Prime $\zeta$ Functions
The problem arose, while I was looking at products of power prime zeta functions
$$
P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks},
$$
with $k\in \mathbb{N}$ and $s=it$ with real $t$.
By ...
9
votes
2answers
540 views
An entire function is identically zero?
I'm preparing for a PhD prelim in Complex Analysis, and I encountered this question from an old PhD prelim:
Suppose $f(z)$ is an entire function such that $|f(z)| \leq \log(1+|z|) \forall z$. Show ...
7
votes
4answers
444 views
Sheaves and complex analysis
A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or ...
5
votes
2answers
223 views
Maximum of sum of finite modulus of analytic function.
Let $f_1,f_2,\ldots,f_n $ be analytic complex functions in domain $D$. and $f = \sum_{k=1}^n|f_k|$ is not constant.
Can I show the maximum of $f$ only appears on boundary of $D\,$?
