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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3
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1answer
90 views

Show that a complex expression is smaller than one

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains ...
2
votes
2answers
115 views

If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $

Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
1
vote
1answer
138 views

Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula

We have the relationship $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ Now I know one uses the Poisson summation formula to prove this. The Poisson summation formula comes from Fourier Transform ...
1
vote
1answer
51 views

Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$

I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$, then $$\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$ When $f(z_0)\neq 0$ and since $1/g$ has a ...
1
vote
1answer
76 views

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$

Show that the equation, $x^3+10x^2-100x+1729=0$ has at least one complex root $z$ such that $|z|>12.$
1
vote
2answers
146 views

Singularity of $f(z)=\frac {\sin z}{e^{-z}+z-1}$

What's the kind of singularity $z_0=0$ of $f(z)$: $$f(z)=\frac {\sin z}{e^{-z}+z-1}$$ I know I should write it in the form: $$f(z)=\frac{\phi(z)}{(z-z_0)^n}$$ but I'm unable to do it.
1
vote
1answer
339 views

The meaning of notation $\subset\subset$ in complex analysis

I have read the book Function Theory of Several Complex Variables of Krantz. But there is a notation the meaning of which I don't know. The notation is $\subset\subset$. For example, "let ...
0
votes
2answers
135 views

Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$

Let $[z_1,z_2,z_3,z_4]$ denote the cross ratio of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized ...
0
votes
1answer
213 views

f is differentiable on convex domain D and Re(f')>0 implies f is injective on D

Suppose that $f$ is differentiable on a convex domain (open and connected) D and Re($f')>0$ in D. How can we prove that f is injective on D? I found some answers using the mean value theorem for ...
0
votes
2answers
347 views

Möbius transformations on $D$ such that $f(D)=D$

I need to find all Möbius transformations on unit disk such that $f(D)=D$, please help!
351
votes
7answers
13k views

“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 $f_n(x)$ ...
54
votes
7answers
11k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
25
votes
4answers
1k views

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$

Hi I am trying to show$$ I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}. $$ Thank you. What a desirable thing to want to prove! It is a ...
21
votes
3answers
1k views

$f^3 + g^3=1$ for two meromorphic functions

Can you find two non-constant meromorphic functions $f,g$ such that $f^3 +g^3=1$?
20
votes
2answers
3k views

on the boundary of analytic functions

Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| < 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the ...
12
votes
6answers
1k 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 ...
21
votes
1answer
353 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} ...
11
votes
2answers
339 views

Why does $\int_0^\infty\frac{\ln (1+x)}{\ln^2(x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
15
votes
5answers
539 views

Integral $\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16\sqrt 2}$

This integral below $$ I:=\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16 \sqrt 2} $$ is what I am trying to prove. Thanks. We can not expand the denominator as a series since the ...
18
votes
4answers
802 views

Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$

I know that $$\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$$ For substituting $u=2$ into $$\zeta(u)\Gamma(u)=\int^\infty_0\frac{x^{u-1}}{e^x-1}dx$$ However, I suspect that there is an easier proof, ...
14
votes
3answers
2k views

Complex Analysis Question from Stein

The question is #$14$ from Chapter $2$ in Stein and Shakarchi's text Complex Analysis: Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_0$ on ...
8
votes
1answer
598 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
10
votes
6answers
5k views

Can we characterize the Möbius transformations that maps the unit circle into itself?

The Möbius transformations are the maps of the form $$ f(z)= \frac{az+b}{cz+d}.$$ Can we characterize the Möbius transformations that map the unit circle into itself?
5
votes
5answers
23k views

Proof of Cauchy Riemann Equations in Polar Coordinates

How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? I ...
14
votes
2answers
408 views

Why can't I combine complex powers

I came across this 'paradox' - $$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i \cdot 2\pi i}=e^{-4\pi^2}$$ I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why ...
11
votes
4answers
2k views

Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.

Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$. My try : I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is ...
11
votes
3answers
2k views

Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
11
votes
2answers
5k views

Branch cut for $\sqrt{1-z^2}$ - Can I use the branch cut of $\sqrt{z}$?

I was trying to clarify some questions I had about elliptic integrals using http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf There they define the map $$\phi\colon w\mapsto ...
10
votes
2answers
403 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. Specifically, ...
9
votes
2answers
934 views

Show that this entire function is polynomial.

Let $f$ be an entire function such that $ |f(z)| \to \infty$ as $|z| \to \infty$. Prove that $f$ is a polynomial.
7
votes
2answers
621 views

How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$

Today I have encounter a series: $$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$ where $u \not \in \Bbb{Z}$ . I have known a method to computer it (by Residue formula): ...
5
votes
2answers
312 views

which of the following is/are true for the entire function $f$?

Let , $f$ be an entire function. Let, $g(z)=\overline{f(\bar z)}$. Let, $D=\{z:Im(z)=0\}\cup\{z:Im(z)=a\}$ for some $a>0$. Then which are correct ? (A) If $f(z)\in \mathbb R$ for all $z\in \mathbb ...
3
votes
1answer
441 views

Modification of Schwarz-Christoffel integral

I found two different formulations of the Schwarz-Christoffel formula (e.g. Link1, p.20 and Link2, p. 9). The first is \begin{align*} ...
12
votes
3answers
712 views

What contour should be used to evaluate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$

Could anyone help me decide what contour to use to evaluate this integral? $$\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$$ So we have simple poles at $i$,$-i$. Why does using a quarter of a circle in ...
11
votes
2answers
612 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 $f(0)=0$. ...
10
votes
1answer
217 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
9
votes
2answers
503 views

Problem on exponential of entire function

I am stuck at a problem which says : " Let $f$ and $g$ are entire functions such that $e^f, e^g$ and $1$ are linearly dependant over $\mathbb{C}$. Show that $f$, $g$ and 1 are also linearly ...
7
votes
1answer
2k views

$\lambda-z-e^{-z}=0$ has one solution in the right half plane

Let $\lambda > 1$ , want to show that the equation $$\lambda-z-e^{-z}=0$$ has exactly one solution in the right half plane $\{z:Re(z)>0\}$. Moreover, the solution must be real.I tried to use ...
6
votes
1answer
5k views

How is Cauchy's estimate derived?

Cauchy's integral formula says $$ f^{(n)}(z)=\frac{n!}{2\pi i}\int_C\frac{f(\zeta)d\zeta}{(\zeta-z)^{n+1}}. $$ If we let $C$ be the circle of radius $r$, such that $|f(\zeta)|\leq M$ on $C$, then ...
5
votes
2answers
2k views

The existence of analytical branch of the logarithm of a holomorphic function

$\Omega$ is a convex open set in $\mathbb {C^n}$ and $f$ is an analytical function Edit: without zero point on $\Omega$, then can we define an analytical branch of $\ln {f}$ on $\Omega$ ?
4
votes
1answer
3k views

Laplace transform of the Bessel function of the first kind

I can't figure out why my evaluation of $\displaystyle \int_{0}^{\infty} J_{n}(bx) e^{-ax} \ dx \ (a,b >0, \ n=0,1,2, \ldots)$ is off by a factor of $ \displaystyle \frac{1}{b}$. $$ \begin{align} ...
2
votes
1answer
105 views

A Tough Problem about Residue

I tried my best to solve this problem from what I learned in residues, but the solution seems very far from what I was doing!! Is there any way other than using Laurent series expansion? Here is the ...
2
votes
3answers
9k views

How to find a Laurent Series for this function

How do I give a Laurent Series on various ranges of $|z|$? I need to find the Laurent series expansion for $$f(z)=\frac{1}{z(z-1)(z-2)}$$ for the following ranges of $|z|$: $0<|z|<1$ ...
13
votes
1answer
277 views

Showing that $2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\left( \frac{x^{a-1}}{\sinh x} - x^{a-2} \right) \, dx$

I want to show that $$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{a-1}}{\sinh x} - x^{a-2}\Big) \, dx \ , \ {\color{red}{-1}} <\text{Re}(a) <1. ...
12
votes
2answers
8k views

Does the complex conjugate of an integral equal the integral of the conjugate?

Let $f$ be a complex valued function of a complex variable. Does $$ \overline{\int f(z) dz} = \int \overline{f(z)}dz \text{ ?} $$ If $f$ is a function of a real variable, the answer is yes as $$ \int ...
9
votes
1answer
721 views

Does there exist an holomorphic function such that $|f(z)|\geq \frac{1}{\sqrt|z|}$?

I have some trouble solving this problem: Does there exist an holomorphic function $f$ on $\mathbb C\setminus \{0\}$ such that $|f(z)|\geq \frac{1}{\sqrt|z|}$ for all $z\in\mathbb C \setminus ...
8
votes
2answers
4k views

Prove that the composition of differentiable functions is differentiable.

Prove that the composition of differentiable functions is differentiable. That is, if $f$ is differentiable at $z$, and if $g$ is differentiable at $f (z)$, then $g\circ f$ is differentiable at $z$. ...
8
votes
2answers
970 views

Characterize entire functions $f$ such that $|f(z)| \leq |\sin(z)|$ [duplicate]

I want to determine all entire functions $f$ such that $|f(z)| \leq |\sin(z)|$. I searched around on MathSEx and I found the following question from which I tried to get inspired but I think it ...
7
votes
2answers
1k views

complex conjugates of holomorphic functions

I came across this question whilst doing some research into complex analysis, and I just can't see what to do! Let $f(z)$ be a holomorphic function on $\mathbb{C}$. Show that ...
7
votes
3answers
510 views

$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic

Prove that the functions $f(z)$ and $\overline{f(\overline{z})}$ are simultaneously holomorphic. I take this to mean that $f(z)$ is holomorphic if and only if $\overline{f(\overline{z})}$ is ...