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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2
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1answer
35 views

Finding the inverse laplace transform using complex analysis.

I've been able to prove simple laplace transforms like $\dfrac {1}{(s+a)} $ quite easily but what about $\dfrac {1}{(s+a)^3+b^2} $ this does not seem easy to do since you cannot easily compute the ...
3
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2answers
57 views

Showing Complex Function is Constant

I am preparing for qualifying exams, and this is a question from the Penn State Qualifying Exam for Fall 2015. It is stated as follows Let $\epsilon > 0$ and let $f$ be holomorphic (analytic) on ...
0
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1answer
26 views

Define a meromorphic function $F$ on $\mathbb C$

Define a meromorphic function $F$ on $\mathbb C$ such that $\displaystyle F(z)=\frac{1}{(z-1)(z-2)} $ when $\Im (z)>0$. That is I want to define $F(z)$ in $\Im(z)\le 0$ in such a way that $F$ is ...
0
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1answer
49 views

Calculate with roots of unity

I have extremly problems to calculate expressions with roots of unity. How do I for example calculate something like this without the use of a calculator: $$a=\frac{(e^{-i\frac{\pi}{4})^2}}{(e^{i\...
3
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3answers
93 views

Complex integral with Residues Theorem

I've been going crazy with this complex integral I have to estimate with the Residues Theorem. I'm obviously missing a sign or something else, but I fear I may be commiting a conceptual mistake. $\...
1
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1answer
58 views

Two tori $\mathbb C/L$ and $\mathbb C/L'$ are isomorph if $L=L'$

Let be two lattices $L$ and $L'$ given such that $L\subseteq L'$. Consider the canonical map from $\mathbb C/L$ to $\mathbb C/L'$. Now I want to show that this map is an isomorphism (biholomorph) if ...
2
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2answers
107 views

Is it natural that $\overline{\int f}=\int\bar f$?

Is it natural that $$\overline{\int f}=\int\bar f\ \ ?$$ I tried to prove it, but with no success.
2
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1answer
30 views

Classifying singularities of $\frac{\sin(\pi z)}{z^4+1}$

If $f(z)=\frac{\sin(\pi z)}{z^4+1}$, we have four roots of unity, which are isolated singularities of $f$: $$z=-(-1)^{1/4},z=(-1)^{1/4}, z=-(-1)^{3/4}, z=(-1)^{3/4}.$$ Do we need to find the Laurent ...
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2answers
25 views

Laurent series of $\frac{1}{z^2(z-1)}$ when $0<\lvert z\rvert<1$

$\frac{1}{z^2(z-1)} = -\left(\frac{1}{z}+\frac{1}{z^2}+\frac{1}{1-z}\right)$. I know that $\frac{1}{1-z}=\sum\limits_{n=0}^\infty z^n$, but what about the other two terms, should they be left as they ...
0
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0answers
19 views

Proof of analyticity in Lewy's Example:

I was reading: https://people.maths.ox.ac.uk/trefethen/pdectb/lewy2.pdf where it is stated: The Lewy Operator on a function $f(x,y,t) : x,y,t \in \mathbb{C}$ is given by $$ \frac{\partial f }{\...
5
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1answer
75 views

Image of circle under fractional linear transform increases in radius

Let $\alpha,r\in\mathbb{R}$ with $r>0$ and $|\alpha|+r\le 1$, and consider the fractional linear transform $$ f(z) = \frac{z-\alpha}{1-\alpha z}. $$ I would like to show the following: the circle ...
0
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3answers
64 views

Finding $n$th derivative in an unusual way

If $f(z) = \frac{e^{iz}}{z^2-1}$, then $f^{(4)}(z)$ can be found by differentiating $f(z)$ four times. I tried to use Cauchy's integral formula, but the integrand is not holomorphic at $z=0$, so we ...
1
vote
1answer
23 views

laurent series expansion of terms like $\frac{1}{z}$, $\frac{1}{z^2}$

I have a homework question of finding the Laurent expansion of $\frac{1}{z^2(z-1)}$ on $0<|z|<1$. I've learned to decompose the function to $\frac{A}{z}$, $\frac{B}{z^2}$, and$\frac{C}{z-1}$, ...
0
votes
1answer
25 views

Finding Taylor series without using derivatives

If $\displaystyle f(z) = \frac{e^{iz}}{z^2-1}$ then we can set $g(z)=e^{iz}$ and $h(z)=z^2-1$. The Maclaurin expansion for $e^{iz}$ is $$\sum\limits_{n=0}^\infty \frac{(iz)^n}{n!}$$ so $\displaystyle ...
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0answers
19 views

Is it possible and interesting to define a complex measure (of curves) based on integrating a meromorphic/holomorphic function?

If we have $f$ a complex function, then we can define $\mu$ like this: given a curve $C$, $\mu(C)= \int_{C} f dz$ I'm suspecting strongly that, unless $f$ is zero a.e or something similar, my ...
0
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1answer
49 views

How can I show that $\left\lvert\sin z\right\rvert^2= \left\lvert\sin x\right\rvert^2 + \left\lvert\sinh y\right\rvert^2$ for $z= x+iy$

I want to show that $\left\lvert\sin z\right\rvert^2= \left\lvert\sin x\right\rvert^2 + \left\lvert\sinh y\right\rvert^2$ for $z= x+iy$ We have that \begin{align} \left\lvert\sin z\right\rvert^2 &...
1
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0answers
17 views

Periods and Dual-Periods of Riemann Surface?

I'm unclear on to what extent the periods and dual-periods of a Riemann surface determine the complex structure of the surface. Perhaps to take a nice example, I'll consider a hyperelliptic curve $y^{...
3
votes
0answers
30 views

Asymptotic behavior of a function defined via a complex integral

I would appreciate any comment/correction about what I did for the following problem, I would be very thankful if you let me know the parts of it which may not be very precise: Let $g(z)$ be defined ...
0
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2answers
47 views

Inequality with complex root and positive imaginary part

Let $z$ be a complex number with $\mathrm{Im}(z)>0$, and we consider $$w:=\frac{-z+\sqrt{z^2-4}}{2}.$$ It is written that "we take the square root so that $\mathrm{Im}(w)>0".$ I want to prove ...
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2answers
50 views

Confusion about singularity of $\sin z$

Consider $f(z)=\sin z$ $(z\not=1)$. I'm confused about the data of my book which I've read. My book says that: "$f$ has removable singularity at $z=1$". My question: how it is possible? I know $\sin ...
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1answer
39 views

$f(z) = \frac{a_0+a_1z^1+…+a_{n-1} z^{n-1}}{b_0+b_1z^1+…+b_n z^n}, b_n \neq 0$. Show that sum of residues at its poles is $\frac{a_{n-1}}{b_n}$ [closed]

Assume that the zeroes of the denominator are simple. Show that the sum of residues at its poles is equal to $\frac{a_{n-1}}{b_n}$ Hi everyone, Iam not able to start the above problem. Hint needed. ...
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0answers
17 views

Contour integral of natural logarithm as part of inverse z-transform

I am trying to apply the inverse z-transform in order to derive a finite difference equation for the following function: $$ H(z) = \frac {1} {ln(z) + a} $$ I can get so far but I am stuck on the ...
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2answers
35 views

To show $f$ conatined in oval is constant

Let $\Omega \subset \mathbb{C}$ is connected open and let $f\in O(\Omega)$ Suppose $f(\Omega) \subset L$ where $L:=\{x+iy \in \mathbb{C} \vert x^{2k} + y^{2k} = 1 \}$ for $k \in \mathbb{Z}^{+}, k >...
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0answers
24 views

Evaluating line integral $I=\int_{\gamma} \frac{z^k}{(z-1)^k}dz$

$I=\int_{\gamma} \frac{z^k}{(z-1)^k}dz$ where $\gamma(t) = 2cos(t)e^{it}$ where $t\in [0,2\pi]$ My attempt : This is a path which is a boundary of $D(1,1)$ traversed twice CCW. thus we get by Cauchy'...
0
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1answer
37 views

Holomorphic branch of square root of $f$

Let $f(x) = (z-\frac{1}{z})$ , $z \in \mathbb{C}$\ {${0}$} Let $F$ be a holomorphic branch of the square root of $f$ that is defined at $z=2$ and has the value $\sqrt{3/2}$ there. GIve an explicit ...
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1answer
29 views

Laurent's Theorem - definition of a path in an annular domain

Theorem on p. 197 of the book mentioned below: Supposed that a function $f$ is analytic throughout an annular domain $R_1<\lvert z-z_0\rvert< R_2$, cenetered at $z_0$, and let $C$ denote any ...
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0answers
34 views

Request for Introduction to Complex valued PDE's and their applications in math and physics

On Mathematica.stackexchange there is a recent question concerning PDE's with complex damping coefficients, see http://mathematica.stackexchange.com/questions/119870/can-i-use-operators-of-the-form-...
1
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1answer
55 views

Normal Convergence in Unit Disk

As I'm preparing for my qualifying exams, I have been given a question, and I'm not sure how to interpret what is being asked. The text I am using is Complex Analysis by Freitag (although the prompt ...
0
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0answers
37 views

Asymptotics of Inverse Laplace transform of a function with a branch point and singularities

consider the inverse Laplace transform $f(x)=L^{-1}[\tilde{f}]$ of a function $\tilde{f}(s)$. I would really like to find the large-$x$ asymptotics of $f(x)$ for the following case: $$\tilde{f}(s)=\...
1
vote
2answers
42 views

Problem: Using Cauchy's integral formula show…

I've stumbled upon a problem I can not solve in the book Mathematical Methods for Electrical Engineers by Thomas B.A. Senior(Page 171). The book gives the following instruction: Using Cauchy's ...
0
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0answers
11 views

A continuity in norm where $p^k\in R^n_{++} $ and $p\in R^n_{+}$

Suppose $X\subset R^n$ is convex and compact and the vector $0\notin X$. Let $p\in R^n_{++}$ and $\langle x,y\rangle_p:=\sum_{i=1}^n p_ix_iy_i$, and $z=\arg\min_{x\in X}\|x\|_p$. Let $$\{x:\...
0
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1answer
29 views

About the proof of Cauchy-Goursat theorem

I am learning complex analysis and encountered a standard proof of the Cauchy-Goursat theorem: if a function f is analytic on a simple closed contour C and its interior then the contour integral of f ...
0
votes
1answer
36 views

Find the harmonic conjugate of $u(x,y)=x^3+ax^2y+bxy^2+2y^3$.

I am trying to find the harmonic conjugate of $u(x,y)=x^3+ax^2y+bxy^2+2y^3$. Using the C-R equations $u_x=v_y$ and $u_y=-v_x$, I have found that $v(x,y)=\frac{b}{3}y^3-\frac{a}{3}x^3-bx^2y-6xy^2+c$. ...
0
votes
0answers
21 views

What are the details of this proof of the open mapping theorem?

here is a proof about which I have some questions: Let $f$ be a non-constant, analytical function on a domain $D$. Let $a \in D$ with $f(a)=b$. Furthermore let the order of $f(z)-b$ at $a$ be n. Then ...
0
votes
1answer
32 views

How to use the completeness here

Suppose that $G\subset \mathbb{C}$ is an open subset and that $X$ is a Banach space. Fix $z_0 \in G$ and let $V:=\{(h,k) \in \mathbb{C}^*\times\mathbb{C}^* : z_0+h \in G\ ; z_0 + k \in G\}$. Let $f: G ...
2
votes
2answers
86 views

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the ...
2
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1answer
52 views

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway's book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic ...
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0answers
54 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. It seems that we can prove it using the winding number with respect to P. ...
4
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2answers
45 views

Singularities of $f(z) = \frac{e^z - e}{(z^2 - 2z + 1)} + z^3\sin(\frac1z).$

In the solution section of my textbook it is said that the function $$f(z) = \frac{e^z - e}{(z^2 - 2z + 1)} + z^3\sin(\frac1z).$$ has a pole of order $1$ at $z_0 = 1$. I don't understand why this is ...
0
votes
1answer
47 views

Triangle inside a simply connected open subset of the complex plane

Let $U$ be a connected open subset of the complex plane. Suppose $U$ is simply connected, i.e. its fundamental group is trivial. Let $T$ be a triangle whose boundary is contained in $U$. It is ...
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4answers
115 views

Find $\int_{0}^{\infty} \sin x^{2}\,dx$

This problem is from my textbook of complex analysis. I have attempted this as: let $$u=x^{2}$$ then $$dx=\frac{du}{2\sqrt{u}}$$ therefore $$\frac{1}{2}\int_{0}^{\infty} \frac{\sin u}{\sqrt{u}}\,du $$ ...
2
votes
3answers
71 views

How do I evaluate this integral using cauchy's residue theorem.

$$\int_0^{2\pi} \dfrac{\cos 2 \theta}{1+\sin^2 \theta}d\theta$$ $$=\dfrac{-2}{i}\oint_{|z|=1} \dfrac{z^4+1}{z(z^4-4z^2-2z+1)}dz $$ I am stuck on how to use Cauchy's residue theorem since the bottom ...
2
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1answer
34 views

Bounded holomorphic and bounded biholomorphic

We call an open subset $U\subseteq \mathbb{C}$ a bounded holomorphic domain, if there exists a non-constant bounded holomorphic function $f:U\to \mathbb{C}$ and we call an open subset $U\subseteq \...
2
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2answers
28 views

Singularities of $f(z) = \frac{z\cos(z) - z}{\sin^3(z)}$

I'm having some difficulties classifying the singularities of $$f(z) = \frac{z\cos(z) - z}{\sin^3(z)}.$$ Here's my work so far: Using the trigonometric identity $\sin^2(z) = (1-\cos^2(z))$ is is ...
2
votes
1answer
44 views

Ways to justify this interchange of summation and integration

In evaluating this integral: $$\int_0^\infty \frac{\Im{\left(e^{e^{ix}} \right)}}{x}\text{d}x$$ My means of evaluation was to expand the numerator of the integrand as a fourier series (a.k.a. Taylor ...
1
vote
1answer
30 views

Find non trivial estimation for a Dirichlet series

I would like to estimate a Dirichlet series, for this I need a estimation for $\sup_{k \in \mathbb{N}}\left|e^{iu f(p^{k+1})}-e^{iu(1+f(p^{k}))}\right|$ where $f$ is real arithmetic addtive function ...
0
votes
0answers
50 views

Compute $\int_0^\infty \frac{\sin(x)}{x}dx$ without residue theorem. [duplicate]

Is there a way to compute $$\int_0^\infty \frac{\sin(x)}{x}dx$$ without residue theorem ?
0
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1answer
37 views

Application of complex analysis and contour integral in generating functions

Normally generating functions are tools of discrete mathematics and integrals deal with continuous structures. A book offered the following formula without much explanation and I'm not able to ...
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21 views

Cauchy-Goursat and removable singularity

Cauchy-Goursat theorem (version 1) says: If $U \subseteq \mathbb{C}$ open set, and $f : U \to \mathbb{C}$ holomorph, then $\int_{\partial T} f = 0 \ \forall T \subseteq U$ Cauchy-Goursat theorem (...
1
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1answer
26 views

holomorph vs having a primitive

Let $U \subseteq \mathbb{C}$ be an open set, and $f : U \to \mathbb{C}$ I know that if $f$ is holomorph, then $f'$ is holomorph, and so derivatives of all orders exists. I know that beeing holomorph ...