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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78
votes
8answers
14k views

Why does $1+2+3+\dots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
215
votes
21answers
19k 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 ...
64
votes
17answers
13k 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 ...
32
votes
10answers
12k 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.
0
votes
2answers
1k 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.
52
votes
3answers
5k 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 ...
4
votes
3answers
585 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 ...
22
votes
1answer
615 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 ...
43
votes
23answers
15k 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 ...
9
votes
4answers
952 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 ...
5
votes
3answers
1k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
18
votes
5answers
1k 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 ...
18
votes
5answers
3k 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.
9
votes
1answer
741 views

Zeta function zeros and analytic continuation

I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
2
votes
3answers
682 views

Difficulties performing Laurent Series expansions to determine Residues

The following problems are from Brown and Churchill's Complex Variables, 8ed. From §71 concerning Residues and Poles, problem #1d: Determine the residue at $z = 0$ of the function ...
2
votes
5answers
455 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:: ...
10
votes
3answers
697 views

A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$

The following question comes from Some integral with sine post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of ...
10
votes
6answers
11k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
12
votes
2answers
732 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 ...
4
votes
2answers
358 views

Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$

For a constant $0<A<\pi$, and natural $n$ I want to find the principal value of the integral: $$\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$$ First of all, I'm not certain what function in ...
12
votes
2answers
1k 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 ...
1
vote
3answers
213 views

How do I compute $\int_{-\infty}^\infty e^{-\frac{x^2}{2t}} e^{-ikx} \, \mathrm dx$ for $t \in \mathbb{R}_{>0}$ and $k \in \mathbb{R}$?

Let $t \in \mathbb{R}_{>0}$ and $k \in \mathbb{R}$. I want to find $$\int_{-\infty}^\infty e^{-\frac{x^2}{2t}} e^{-ikx} \, \mathrm dx.$$ A hint told me to first determine $\int_{-\infty}^\infty ...
15
votes
2answers
804 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 ...
0
votes
1answer
174 views

For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?

For example, when $\alpha = 2$, $\ln(z^{2}) \neq 2\ln(z)$, because argument z is determined up to constant $2 \pi k$. So $$ \ln(z^{2}) = \ln(z) + \ln(z) = \ln(z_{k_{1}}) + \ln(z_{k_{2}}) \neq ...
34
votes
19answers
4k 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 ...
4
votes
3answers
2k views

Entire function bounded by a polynomial is a polynomial

Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
2
votes
1answer
519 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 ...
3
votes
1answer
232 views

Using calculus of residues to evaluate a trig integral

This is my partial attempt at the solution. I am unsure how to proceed further.
6
votes
2answers
979 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. $\frac{z-1}{(z-2)(z-3)}$ = ...
137
votes
4answers
9k 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 ...
17
votes
5answers
1k 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: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
16
votes
1answer
423 views

“Orientation” of $\zeta$ zeroes on the critical line.

I am pretty ignorant about complex analysis so please forgive my lack of terminology. I saw a pretty picture (posted below) of the behavior of the Riemann zeta function along the critical line. What ...
15
votes
1answer
978 views

Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_nz^n+\cdots+a_0$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
5
votes
2answers
703 views

About the limit of the coefficient ratio for a power series over complex numbers

This is my first question in mathSE, hope that it is suitable here! I'm currently self-studying complex analysis using the book by Stein & Shakarchi, and this is one of the exercises (p.67, Q14) ...
4
votes
3answers
392 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?
10
votes
2answers
704 views

Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem

Let $f(z)$ be entire function. Consider the functions $e^{if(z)}$ and $e^{−if(z)}$ and applying the Maximum Modulus Theorem, show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant ...
4
votes
5answers
793 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 ...
1
vote
2answers
292 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? ...
5
votes
2answers
416 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 $D\,$?
3
votes
3answers
348 views

Recursion relation for Euler numbers

I am trying to solve the following: The Euler numbers $E_n$ are defined by the power series expansion $$\frac{1}{\cos z}=\sum_{n=0}^\infty \frac{E_n}{n!}z^n\text{ for }|z|<\pi/2$$ (a) ...
3
votes
3answers
376 views

$|e^a-e^b| \leq |a-b|$

Came across this problem on an old qualifying exam: Let $a$ and $b$ be complex numbers whose real parts are negative or 0. Prove the inequality $|e^a-e^b| \leq |a-b|$. If $f(z)=e^z$ and $z=x+iy$, ...
15
votes
2answers
460 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
12
votes
2answers
870 views

Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform

Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
7
votes
1answer
570 views

Multiple choice question: Let $f$ be an entire function such that $\lim_{|z|\rightarrow\infty}|f(z)|$ = $\infty$.

Let $\displaystyle f$ be an entire function such that $$\lim_{|z|\rightarrow \infty} |f(z)| = \infty .$$ Then, $f(\frac {1}{z})$ has an essential singularity at 0. $f$ cannot be a polynomial. $f$ ...
9
votes
2answers
697 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 $\sin(x) = {e^{ix}-e^{-ix} \over 2i}$? $$ I=\int_0^{\pi/2} \ln ( \sin(x)) \; dx = - {\pi \ln(2) \over 2}$$
6
votes
1answer
303 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 ...
4
votes
4answers
150 views

I am trying to show $\int^\infty_0\frac{\sin(x)}{x}dx=\frac{\pi}{2}$

I am trying to show $\int^\infty_0\frac{\sin(x)}{x}dx=\frac{\pi}{2}$ It was an exercise from a book about complex analysis, so I've gone through the complex plane to do it! Consider a semi-circle ...
11
votes
3answers
3k views

How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
5
votes
2answers
305 views

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
4
votes
1answer
626 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} ...