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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49 views

Calculating complex numbers

What is $(1-i)^{1+i}$ ? I did: $(1-i)^{1+i} = \left(e^{ln(1-i)}\right)^{1+i} = e^{(1+i)\cdot ln(1-i)} = e^{(1+i)\cdot [ ln\sqrt{2} + i\cdot(-\pi/4) + 2ki\pi ]}$ $ = e^{(ln\sqrt{2} + i\cdot(-\pi/4) + ...
1
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0answers
79 views

Complex analysis visualization (Cauchy Theorem, Residue Theorem)?

I usually think of complex functions on the complex plane like vector fields. So basically what I have problems with is visualizing firstly Holomorphic functions. I have also read and successfully ...
12
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3answers
635 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
1
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1answer
109 views

2 holomorphic functions from Unit disk into itself

Suppose $\phi_{1}, \phi_{2} : D -> D$ are 2 holomorphic functions from unit disk into itself such that $\phi_{1}(0) = \phi_{2}(0) = 0$ and $\phi_{1}(D)$ contains $\phi_{2}(D)$. I want to show that ...
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0answers
89 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
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0answers
213 views

Jordan's Lemma for a specific contour integral

I was solving a problem where i need to integrate the complex function : y = $\int_C{\dfrac{e^{z}}{(z+2)^{2}}} dz$ over a contour C where the function $e^{z}\left(\dfrac{z-1}{z+2}\right) \rightarrow ...
30
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3answers
832 views

Why does the graph of $e^{1/z}$ look like a dipole?

I was looking at the color wheel graph of $e^{1/z}$, and my girlfriend commented that it looked just like a dipole. Does anyone have an explanation for that, why the geometry would be so similar? I ...
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1answer
47 views

Calculate Complex Limit

Assume that $W$ is analytic in a domain $D$. Let $z_0$ be fixed with $W'(z_0) \ne 0$ and let $f$ be defined for all $z \in D-{z_0}$, $$f(z) = \frac{W'(z_0)W'(z)}{(W(z)-W(z_0))^2} - ...
1
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1answer
83 views

Is the infinite union of given sets closed or open?

This is a homework question, and i need to determine whether the following union $\cup _{n=1} ^ \infty I_n$ is either an open or a closed set, where $I_n = [0,a_n]$ and $a_n=\cos \frac{\pi}{2n}+i \sin ...
1
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1answer
32 views

Can there exist an FLT mapping UHD to UHP?

Can there exist a fractional linear transformation mapping the upper half-disc conformally onto the upper half-plane? I think there cannot be, because I am looking for where the unit circle and the ...
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1answer
105 views

Evaluating $I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$ for real $a$ and real $n > 0$ with Jordan's Lemma

Given that $e^{-nx^2}$ does not have any singularities, I believe $I(a,n) = 0$. Is this a correct application of Jordan's Lemma? $$I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$$
3
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4answers
207 views

Evaluating $I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx$ for real $n$

I am not sure how to handle the additional parameter $n$. I first need to find out for which real values of $n$ will the integral converge. Based on intuition and checking with mathematica, I believe ...
2
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0answers
84 views

$U\subseteq \mathbb C$ is simply connected if and only if it is connected and $U^c$ has no bounded components?

If we define simply connected sets as connected open sets so that every cycle in them is homologous to $0$ w.r.t. them (has its interior in them) is it true that $U\subseteq \mathbb C$ is simply ...
2
votes
1answer
324 views

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...
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1answer
51 views

Prove that $F_a(z) - F_b(z) = F_a(b)$

We had the following statement. Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
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2answers
63 views

Prove g(x) is constant

I have a question in my class If $g$ is an entire function and $g(z)=g(z+z')$ for all $z\in C$ and $z'=a+bi,a,b\in Z$. How to prove g is constant
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0answers
26 views

How to show $\int_{\sum_{i = 1}^n \gamma_i} f = \sum_{i = 1}^n \int_{\gamma_i} f$

Let $G \subset \mathbb C$ be an open set, $f: G \to \mathbb C$ a continuous function and $\gamma: [\alpha, \beta] \to G$ a contour. Define the contour integral of $f$ along $\gamma$ to be ...
3
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1answer
146 views

Holomorphic self map on the open unit disc

Does there exist a holomorphic function $f:\mathbb{D}\longrightarrow\mathbb{D}$ such that $f(1/2)=-1/2$ and $f'(1/4)=1$, where $\mathbb{D}=\{z:|z|<1\}$?
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1answer
975 views

Mean value theorem for holomorphic functions

The mean value theorem for holomorphic functions states that if $f$ is analytic in $D$ and $a \in D$, then $f(a)$ equals the integral around any circle centered at $a$ divided by $2\pi$. But if $f$ ...
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2answers
62 views

Connected components in the plane

I would greatly appreciate if you could give some reference to my question. There compact set $K$ for which $\mathbb{C}\setminus K$ have to have infinite connected components?
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0answers
73 views

heat equation problem

Consider a rectangular plate, of size $d$ in the $y$ direction and size in the $x$ direction. At $y = d$ the temperature is held fixed at $f(x)$. On all other sides the plate's temperature is held at ...
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1answer
223 views

Complex integration over a general ellipse

I'm having trouble evaluating the complex integral over an ellipse : $\int_C{\dfrac{1}{z^{4} + 1}} dz$ where C is the ellipse given by $x^{2} - xy + y^{2} + x + y = 0$. How should I go about it?
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1answer
254 views

Need clarity with the maximum modulus principle of analytic functions

I was reading on the maximum modulus principle and I stumbled upon a Theorem: If a function $f$ is analytic and not constant in a given domain $D$, then $|f(z)|$ has no maximum value in $D$. That ...
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1answer
166 views

.cauchy integral formula.

$\int \frac{e^{zt}}{\sinh z} \mathrm{d}z$ and $|z|=8$ this is the problem about Cauchy integral formula. I keep making a mistake and find the wrong solution. Can you help me? So, we know the Cauchy ...
4
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1answer
194 views

Conformal mapping of disk, surjective, not injective

Is there an example of a conformal mapping of the disk onto itself which is not injective? If not, how may we prove there does not exist such a map? This came up in the answer to this question.
3
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2answers
57 views

Show $\left|\int_\alpha^\beta F(t) dt\right| \le \int_\alpha^\beta |F(t)| dt$

Let $F: [\alpha,\beta] \to \mathbb C$ be a continuous function, $F(t) = u(t)+iv(t)$. Define the integral of $F$ over $[\alpha,\beta]$ to be \begin{align*} \int_\alpha^\beta F(t) dt = ...
4
votes
2answers
828 views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
17
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3answers
370 views

Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone ...
1
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1answer
119 views

how do you convert y" equation into Sturm-Liouville form

For $y = y(x)$, convert the following equation $$ y''- 2xy' + 2vy = 0;$$ where $v$ is a constant, into a Sturm-Liouville form $$ Ly = r(x)(\lambda)y,$$ $\lambda $ is a number, where $$ L := ...
5
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2answers
2k views

Integrating $\int_0^\infty \frac{1-\cos x }{x^2}dx$ via contour integral.

In Stein's Complex Analysis notes, the following exampleis given. They then proceed to calculate the integral over the small semicircle. My question is, why is it necessary to dodge the origin? ...
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1answer
55 views

Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
4
votes
3answers
93 views

$\lim_{h\rightarrow 0} \dfrac {e^{f(z+h)}-e^{f(z)}}{f(z+h)- f(z)}$

given that $V$ is an open subset of $\mathbb{C}$ and $z \in V$, calculate $\lim_{h\rightarrow 0} \dfrac {e^{f(z+h)}-e^{f(z)}}{f(z+h)- f(z)}$, if $f$ is known to be a continuous complex function in ...
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0answers
58 views

How do you integrate the “logarithmic part”?

This entry in Wikipedia states the following theorem (http://en.wikipedia.org/wiki/Partial_fraction_decomposition#Application_to_symbolic_integration) Let $f$ and $g$ be nonzero polynomials over a ...
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votes
2answers
298 views

simple tools to extract Re,Im,Abs… of any complex function

I've developped my own set of simple yet powerful tools to work on complex functions. I would like to know if these simple tools are currently used in complex analysis. Let's $z = x + i y = |z| ...
5
votes
3answers
129 views

How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$?

If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4 = 0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + ...
4
votes
1answer
118 views

Which conformal maps UHP$\to$UHP extend continuously to the closure?

Does every conformal map of the upper half-plane $\{\text{Im }z >0\}$onto itself extend continuously to a map from its closure $\{\text{Im }z \geq 0\}$ to itself? If not, which ones do? In ...
1
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1answer
28 views

Give the region of validity.

Let $F(z)=\log(z^3-8)$. Let $z=re^{i\theta} \Rightarrow z^3=r^3e^{i3\theta}=8 \Rightarrow r=2$ and $\theta=\frac{2\pi k}{3}$ for $k=0,1,2,3$. So, $F$ is analytic on ...
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2answers
71 views

Expand $sin(z)$ about $\pi$

It seems like all you have to do is use the fact that: $f(z)=\sum_{n=0}^\infty\frac{f^n(z_o)}{n!}(z-z_o)^n$ In this scenario, $z_o=\pi$. The solution I got is: $\frac{sin(\pi)}{0!}(z-z_o)^0 + ...
3
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3answers
109 views

Solving equation system of complex funtions

Does there exist two complex functions $f$ and $g$ satisfy below equation system? $$ \begin{cases} f=e^g\\ g=e^f \end{cases} $$ What about analytic funtions?
6
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1answer
106 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
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0answers
34 views

Give an appropriate region on which the function is analytic.

$F(z)=\log(z^4+4i)$ Here is what I did, Let $z=re^{i\theta} \Rightarrow z^4 =r^4e^{i4\theta}=-4i \Rightarrow r^4=4, \theta=\frac{\frac{-\pi}{2}+2\pi k}{4}$ for $k=0,1,2,3$. Now $F$ is analytic on ...
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0answers
32 views

Dirichlet series minus Riemann zeta

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Consider ...
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1answer
58 views

How to prove this?.

let $f:D\to D$ be a holomorphic function on the unity disk $D$. If $f(0)=0$, prove that $|f'(0)|\le1$.
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1answer
32 views

Special polynomials having atleast one root on the unit circle

I have the following problem: For each $w\in\mathbb{T},$ ($\mathbb{T}$ denotes the unit circle), consider the polynomial $P_{w,n}(z)=z^{n+1}+z^n-2w$ of degree $n+1,$ where $n\in\mathbb{N}.$ Does there ...
1
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0answers
33 views

How does $e^{az}$ change when $z$ is shifted $2i\pi$?

I want to evaluate an real-valued integral using residue theory, and the approach is to shift the fraction $\displaystyle\frac{e^{ax}}{1+e^x}$ from the real line up $2i\pi$ in the complex plane and ...
2
votes
1answer
124 views

Entire Function

I am facing the following problem. Let $f$ and $g$ be analytic functions in $|z|<1$, with $$f(z)=\sum_{n=0}^{\infty}a_nz^n,\quad g(z)=\sum_{n=0}^{\infty}b_nz^n$$ such that $a_n\geq 0$, $b_n\geq 0$ ...
4
votes
1answer
111 views

The sheaf $\mathfrak{S}$ of germs of analytic functions over $D$ is a topological group (Ahlfors)

In Ahlfors' complex analysis text, page 286 he gives the following definition: Definition 1. A sheaf over $D$ is a topological space $\mathfrak S$ and a mapping $\pi:\mathfrak S \to D$ with the ...
2
votes
1answer
71 views

Analytic function or not?

Is $f(t) = 1 + e^{2\pi i \phi t}$ a complex analytic function? $t\in\mathbb{R},\phi\in\mathbb{Z},i=\sqrt{-1}$. I know this could be an easy question, but I just want to make sure that the 1 does not ...
0
votes
1answer
54 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
1
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1answer
29 views

Prove that $f(x)=(e^{ix}-e^{iz_0})f_1(x)$ where $f_1(x)$ is also a trigonometric polynomial

Let $f(x)=\sum_n c_ne^{inx}$ be a trigonometric polynomial. It then makes sense to define $f$ on $\mathbb{C}$ by allowing $x$ in this formula to be any complex number. Suppose $f(z_0)=0$ for some ...