# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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$\frac{C}{z-1}$, ...
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### 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 ...
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### 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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### 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'...
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### 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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### 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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### 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-...
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### 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 ...
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### 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 ...
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### 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 $v(x,y)=\frac{b}{3}y^3-\frac{a}{3}x^3-bx^2y-6xy^2+c$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 ...
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 (...
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 ...