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, ...

learn more… | top users | synonyms (2)

0
votes
0answers
3 views

Isolated singularity of $\frac{\sin(\pi z)}{z - 1}$

I'm learning complex analysis, specifically Laurent series and isolated singularities, and need help to understand the solution to the following exercise: Find and determine the nature of the ...
0
votes
0answers
9 views

Theta series and Jacobi theta functions

I have some difficulties with expressing the following series $\sum\limits_{b=-\infty}^{+\infty}\sum\limits_{a=-\infty}^{+\infty}q^{1 + 3 a + 3 a^2 - 3 b - 3 a b + 3 b^2}$ using standart theta ...
1
vote
2answers
35 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
2
votes
0answers
36 views

Let $f:\mathbb{C} \to \mathbb{C}$ be an non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$.

Let $f:\mathbb{C} \to \mathbb{C}$ be a non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$. Then prove that $f$ is surjective. It can be solved trivially by Picard's ...
2
votes
0answers
30 views

Let $f(z)$ be an entire function. Prove that $f$ and $f-a$ have the same order.

Let $f$ be an entire function, the order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r},$$ where $M_{f}(r)=\max_{|z|=r} |f(z)|$. And this is equivalent to define $...
0
votes
1answer
22 views

Construct a function which satisfies the following conditions

I am struggling with the following routine problem : Construct a holomorphic function f(z) with pole of order 2 at 0, an essential singularity at 1 and with residues 1 and 0 respectively. In general ...
0
votes
0answers
31 views

Polynomial with roots on ellipse in complex plane [on hold]

What polynomial or complex equation produces 4 equi-spaced complex numbers Z with one real root $(1,0)$ ( for $\theta =0$) on an ellipse in the complex plane, where $$ Z =\frac{e^{i\theta}}{1-\...
3
votes
1answer
41 views

Is this a contour integral question?

I had this in my previous cats that I'm not sure whether it's really a complex analysis question, looks like a differential question with line integrals a bit $$\int_{(1,3)}^{(4,5)} (2y+x^2)\,dx + (...
2
votes
0answers
26 views

Comparison of entire functions with $f=cg$

Let $f,g:\mathbb C\to\mathbb C$ be holomorphic with $|f(z)|\leq|g(z)|$ for all $z\in\mathbb C$. Show that there is a $c\in\mathbb C$ with $|c|\leq 1$ such that $f=cg$. If $g\equiv 0$ then we ...
9
votes
0answers
189 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\...
2
votes
1answer
40 views

How to derive this Hankel's Contour integral formula with gamma function?

This relation was put up in The Art Of Computer Programming and no derivation was offered. Please help me understand this better. $$\frac{1}{\Gamma (z)} = \frac{1}{2i\pi} \oint\frac{e^t dt}{t^z}$$ ...
2
votes
0answers
34 views

Convergence of a series of complex numbers.

Let $f : \mathbb C \to \mathbb C$ be a non constant entire function. Does the series $\sum_{n=1}^ \infty \frac{1}{n} f(\frac {z}{n})$ converges at any point $z \in \mathbb C$ ? I think this will not ...
1
vote
0answers
34 views

topology of compact convergence, closed sets

Let $H(\mathbb{D})$ be the vector space of all analytic functions on the unit disk. Then the topology induced by uniform convergence on compact subsets is metrizable. Thus the following topology ...
0
votes
0answers
12 views

About some good references for self study

I'm willing to start a self study of Hardy spaces, Bergman spaces and Bloch spaces. I would like to know good books on the subject. Since I'm going to study on my own, would be great to find one that ...
0
votes
2answers
41 views

Argument of complex numbers

If $z=re^{i\theta}$ and $w=\rho e^{i \phi} $ are two complex numbers, then $ arg(zw)=arg (z)+arg (w)$ But if $z=-1$ and $w=-1$, we get $ 0= 2\pi $ which is not correct. So why it gives us this ...
2
votes
0answers
48 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
1
vote
2answers
51 views

Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
1
vote
3answers
110 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
3
votes
1answer
40 views

Show that $F(z) \in H(D_1(0))$

Let $\mathbb{S}^{1} = \{ z \in \mathbb{C} : |z| = 1 \}$ and $f: \mathbb{S}^1 \longrightarrow \mathbb{C}$, $f \in \mathcal{C}^{0}(\mathbb{S}^{1})$, i.e., $f$ is continuous. Define for $z \in D_1(0) = \{...
0
votes
2answers
50 views

$\log(e^z - i)$ as a holomorphic function in $\mathbb{D}$

I'm learning complex analysis, specifically holomorphic functions, and need help with the following exercise: Examine if the function $\log(e^z - i)$ can be defined as a holomorphic function in ...
0
votes
1answer
33 views

Proof of non-constant analytic functions

Use the following theorem: "A function that is analytic in a domain $D$ is uniquely determined over $D$ by its values in a domain, or along a line segment, contained in D" to show that if $f(z)$ ...
1
vote
1answer
23 views

Proof of Reflection Principle when f(x) is imaginary

Suppose that a function f is analytic in some domain $D$ which contains a segment of the x-axis and whose lower half is the reflection of the upper half with respect to that axis then $$\overline{f(z)...
3
votes
2answers
37 views

Conformal holomorphic mapping from disc to square

Let $f$ be a holomorphic map from the unit disc $\mathbb{D}$ to an open square $\mathbb{S}$ with its center at the origin. Given $$ f(0) = 0, \qquad f'(z) \neq 0 \quad (z \in \mathbb{D}) $$ prove that ...
1
vote
1answer
31 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3$ be any point of the space. I intuitively suppose that the Lebesgue integral $$\...
1
vote
1answer
31 views

Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
1
vote
0answers
84 views

Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. Is everything an analyst could ever care about locally homeomorphic to a T1 topological ...
1
vote
1answer
25 views

Is a Blaschke product/rational function a covering map for a $n$-sheeted covering of $S^{1}$?

We have a Blaschke product $B(z)$ of order $n$ (you can think of it as a rational function with $n$ zeros and $n$ poles), the zeros are obviously inside $\mathbb{D}$. Why is $B(z) \colon S^{1} \to S^{...
0
votes
2answers
22 views

Automorphism of Upper half plane

Let $M=\left\{\left.\displaystyle z\mapsto\frac{az+b}{cz+d}\ \right|\ \ ad-bc\not =0\right\}$,$$p:GL(2,\mathbb C)\to M, \begin{bmatrix}a & b \\ c & d \end{bmatrix}\mapsto\frac{az+b}{cz+d}.$$ ...
-1
votes
1answer
23 views

Finding the Laurent series of a function

I'm trying to work through the following example: Find the Laurent series of: $$ f(z) = \frac{1}{z(z-2)^3}, $$ about the singularities $z = 0$ and $z = 2$ (separately). Hence ...
0
votes
0answers
37 views

contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
1
vote
1answer
34 views

Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
0
votes
1answer
20 views

Contour Integral involving hyperbolic functions

I would like to evaluate: $\displaystyle\oint_C \frac{e^{4z}-1}{\cosh(z)-2\sinh(z)}\,\mathrm dz$ where $C$ is a unit circle in the complex plane and $z=x+iy$. I did not find any singular ...
2
votes
1answer
33 views

Does this function Contradict this colloary?

Colloary: If $G$ is a domain and $f: G \to \mathbb{C}$ is analytic and not identically zero, then the zeros of $f$ are isolated. If the domain $G$ is closed and bounded, then the zeros are finite in ...
1
vote
0answers
52 views

Elementary fact about holomorphic function

I found some assertion in this article (bottom of third page) of François Trèves which in a way states that taking a bounded simply connected domain $D$, $H$ the open upper half plane, a holomorphic ...
0
votes
3answers
44 views

How to determine Laurent series associated to $f(z)$ [on hold]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
0
votes
0answers
17 views

Prove that these are the automorphisms of the Poincare upper half plane

If $P = \{Im(z) > 0\} \subset \mathbb{C}$, prove that $f:P \rightarrow P$ is an automorphism iff $f(z) = \frac{az+b}{cz+d}$, for $a,b,c,d \in \mathbb{R}$, with $ad-bc > 0$. I had thought about ...
0
votes
1answer
28 views

Does $f(z)$ has an essential singularity at $z=z_0$?

Let $f(z)=g(z)h(z)$ such that $g$ has an essential singularity in $z=z_0$ and $h$ is holomorphic in a neighbourhood of $z_0$ then $f$ has an essential singularity in $z=z_0$? Im trying to see this ...
6
votes
1answer
106 views

Evaluate $\int \frac {\sin(x)}{x^2 + 4x + 5}dx$

Question: Evaluate $$ \int \frac{\sin(x)}{x^2 + 4x + 5} dx=\int \frac {\sin(x)}{(x + 2)^2 + 1}dx $$ By using the change of variable $y = x + 2$ we have that $dy = dx$ then $$I = \int \frac{\...
-1
votes
1answer
47 views

Residue of a trig function multiplied by a polynomial

can somebody help me to find the residue for: I tried to make two series centered at $(z - k\pi)$ for $\sin(z)$ and $1- \cos(2z)$ but I don't know what to do with the $(z+\pi)^2$....and obviously, i ...
1
vote
1answer
35 views

Write $\,-4i\,$ in polar form

Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$. I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\...
2
votes
4answers
170 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
0
votes
1answer
19 views

square of polynomial still harmonic? [on hold]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
2
votes
1answer
76 views

Sufficient condition for an holomorphic map to be conformal

Let $U,V\subseteq\Bbb C$ be open sets, let $f:U\to\Bbb C$ be holomorphic. If we want to prove that $f$ is a conformal map $U\to V$, my teacher said that is enough to check that $f$ is locally ...
0
votes
3answers
69 views

$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$?

Can we compute the integral ? $$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$$ Actual problem asks to compute: $$\int_{|z|=2} \frac{z^n}{\sqrt{z^2+1}} dz$$ To compute this I need to solve the integral: $\...
10
votes
1answer
3k views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
2
votes
3answers
103 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...