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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65 views

Integral Evaluation Trouble

I have the following indefinite integral: $$\int\frac{2e^x}{e^{2x}+1+2x}dx$$ I was thinking substitution, so let $u=1+2x$. then $du=2dx$ and thus $$\int\frac{e^{\frac{u-1}{2}}}{e^{u-1}+u}du$$ I've ...
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2answers
60 views

Derivatives of $f$ doesn't match accordingly to the Cauchy-Riemann equations?

For the function $f(x+iy) = 2xy+i(x+\frac2 3y^3)$, I've decided that $f$ is differentiable at the points $-1/2$ and $-1/2 + i$ by using the Cauchy-Riemann equations: $\frac {\partial u}{\partial x} = ...
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2answers
27 views

Establishing a bound in the complex plane

so my function is $$f(z)= \frac{e^{iz}}{z^2+a^2} $$ What is getting to me and probably I should've been comfortable with this fact is how they establish this upper bound: $$\bigg|\int_{0}^{\pi} ...
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1answer
29 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
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1answer
45 views

Determine all points where $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$ is differentiable in $\mathbb C$.

Consider the function $f : \mathbb C \rightarrow \mathbb C$ given by $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$. I want to determine all points at which $f$ is differentiable as a complex function. To do ...
5
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1answer
81 views

How to prove $\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $ around a phase singularity/over a cut

How would you prove that $$\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi $$ We know that $\theta\in(-\pi,\pi)$, suppose that $\theta$ is continuous in the region bounded by and along $\Gamma$ apart ...
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1answer
55 views

Show that the meromorphic differential of the homogeneous polynomial is holomorphic and not isomorphic to $\mathbb{P_1}$

Consider the elliptic curve i.e. non-singular cubic, $X$ given by the equation $\xi_0\xi_2^2=\xi_1^3-\xi_0^2\xi_1$ in projective coordinates $(\xi_0:\xi_1:\xi_2)$, or, equivalently, by the equation ...
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1answer
35 views

Evaluate the countour integral $\int _\Gamma z dz$

Can someone please help me setup a) $\int _\Gamma z dz$ b) $\int _\Gamma \bar z dz$ and given the admissible parametrization of $\Gamma$ $\Gamma_1 : z_1: 2 + i(t - 1) ; 1 \leq t \leq 2$ and ...
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0answers
25 views

Fermats of the curve

Consider Fermats curve of degree 4 defined by $x_1^4 + x_2^4 = x_0^4$ in projective coordinates $ (x_0, x_1,x_2)$ . Find the divisor of the meromorphic differential $(w) = dx/y^3 $ on the curve. ...
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1answer
70 views

Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
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2answers
26 views

Show $\\Log z_1z_2 \neq Log z_1 + Log z_2$. given $z_1 = i$ and $z_2 = -\sqrt 3 + i$.

Show by evaluating both sides that for $z_1 = i$ and $z_2 = -\sqrt 3 + i$, $\\Log z_1z_2 \neq Log z_1 + Log z_2$. Recall the definition: $\\Log z = Log |z| + iArg z$ Attempt: left side: $\\Log ...
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1answer
39 views

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close ...
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3answers
110 views

Why does the residue method not work straight out of the box here?

I'm trying to evaluate the integral $$I = \int_0^{\infty} \frac{\cos(x)-1}{x^2}\,\mathrm{d}x $$ The way I've done this is by rewriting $\frac{\cos(x)-1}{x^2}$ as ...
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1answer
40 views

Laurent series of $\frac{1}{\sin^2z}$ around $0$?

I tried to expand $\frac{z^2}{\sin(z)^{2}}$ using Taylor expansion, but the coefficient involved some limit of $\frac{0}{0}$ and was really difficult to calculate. (I tried to convince myself the ...
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2answers
44 views

complex integral using cauchy's formula

I have to find the value of: $\int_{|z-2|=\frac{3}{2}}\frac{\cos{z}}{z^2(z^2-\pi^2)}$ If there is no singularity in the closed disk $D=|z-2|\leq\frac{3}{2}$ then the integral should be 0. For the ...
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2answers
52 views

A dense subspace of L^2

Let $\mathcal{H}$ be the Hilbert space of holomorphic functions defined on the unit disc $D\subset\mathbb{C}$ which is the clousure of the complex polynomial functions on the disc with respect to the ...
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2answers
70 views

Trig function integral

I'm trying to solve $$\int_{0}^{\pi}\frac{dx}{cos^2(x)-a^2}, \hspace{5mm} 0<a<1$$ There are numerous examples of similar integrals but non with the condiction that $0<a<1$, say $a = ...
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1answer
25 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : ...
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1answer
42 views

Suppose $|\alpha + \beta| \le |\alpha' +\beta'|$. Is it then possible that $|\alpha'| + |\beta'| \le |\alpha| + |\beta|$?

Suppose $\alpha, \beta, \alpha', \beta' \in \mathbb C$ and that $|\alpha + \beta| \le |\alpha' +\beta'|$. By using the triangle inequality is it then possible that $|\alpha'| + |\beta'| \le ...
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2answers
32 views

Show that the function $g(z) = f(e ^z )$ is not a polynomial.

Let $f :\mathbb C \rightarrow \mathbb C$ be an entire function. Show that the function $g(z) = f(e ^z )$ is not a polynomial. What is the technique to show a function is a polynomial?
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2answers
75 views

How do I use residue theorem to evaluate this improper integral to get a good looking solution?

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$ I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...
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2answers
39 views

Finding the zeroes of this complex polynomial

Now I thought it wouldn't be too much of an issue, but it is becoming hell to find the zeroes of: $$z^4 + 10z^2 +1 $$ Now reason I need them is for the poles of a function I am working on. So with ...
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0answers
69 views

simple root of a complex variable function

I have encountered the following question: Let $f_{\epsilon}(z)=\sin(z)+\epsilon e^z$. Prove that $f_{\epsilon}(z)$ has a simple root $z_{\epsilon}$ with the property that $\lim_{\epsilon \to 0} ...
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1answer
16 views

How does this inequality of a complex function hold

I cannot figure out how $\Re[g(\lambda)]\leq |\lambda|$ implies $|g(\lambda)|\leq|2 r-g(\lambda)|$ where $\lambda$ is an arbitrary complex number s.t. $|\lambda|\leq r$, and $g$ is an entire function. ...
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1answer
30 views

Residue of 1/sin^3(z)

What are the residues of $ \frac{1}{sin^{3}z} $? From the residue theorem the residues are at $$\lim_{z \rightarrow z_{0}} \frac{1}{(n-1)!} \frac{d^{n-1}}{dz^{n-1}} (z-z_{0})^{n} f(z)$$ $$\lim_{z ...
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1answer
78 views

Find the Laurent expansion for $f(z)=\frac{\exp{1/z^2}}{z-1}$ about $z=0$.

Find the Laurent expansion for $f(z)=\frac{\exp{(1/z^2)}}{z-1}$ about $z=0$. I was able to determine the series for each of the factors. We have ...
2
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1answer
48 views

Maclaurin Series: Complex Analysis

Question: Use the representation $\sin z = \sum\limits_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}$, $|z|<\infty$ to write the Maclaurin series for the function $f(z) = \sin z^2$ and point out how ...
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1answer
30 views

separate the vector into real and imaginary parts

How does one separate below vectors into the proper form? I've tried everything I could think of, including suggested answers below that should work but don't. c +/- di is required form. Thanks. ...
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1answer
15 views

Laurent Series Proof: complex analysis

Question: Show that when $0 < |z - 1| < 2$, $\frac{z}{(z-1)(z-3)} = -3\sum\limits_{n=0}^\infty \frac{(z-1)^n}{2^{n+2}} - \frac{1}{2(z-1)}$ Attempt at solution: My understanding of series in ...
2
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1answer
32 views

Find limit of sequence, complex variable

I need to calculate limit of this sequence: $$\left\{\left(1+\frac{1}{n}\right)\exp\left(\frac{i\pi}{n}\right)\right\}$$ My problem is that I can't extract real and imaginary parts. I think something ...
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0answers
32 views

A contour integral value

While working on a difference equation solution a use of the Laplace transformation was made. The inversion leads to the integral \begin{align} \frac{1}{2\pi i} \int_{\gamma - i \infty}^{\gamma + i ...
4
votes
1answer
86 views

Find the Laurent Expansion of $f(z)$

Find the Laurent Expansion for $$f(z)=\frac{1}{z^4+z^2}$$ about $z=0$. I have found the partial fraction decomposition $$f(z)=\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{2i(z-i)}+\frac{1}{2i(z+i)}.$$ ...
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1answer
48 views

Complex function of class $C^m$

Let $m$ a positive integer and consider the function $$f(z)=\vert z\vert^\alpha z$$ with $\alpha>0$. I have to find the value of $\alpha$ for which $f\in C^m(\mathbb{C},\mathbb{C})$. Now if ...
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0answers
34 views

Show that the function sending $f$ to its derivative is continuos in compact-open topology.

Let $\Omega$ $\subset$ $\mathbb{C}$ be an open subset of the complex plane. We need to show that the function $D$: $\mathcal{H}(\Omega)$$\rightarrow$$\mathcal{H}(\Omega)$ $f \rightarrow f'$ is ...
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0answers
50 views

Complex number as function of real number

While seeking all solutions of $ Z ^ 2 = 2 ^ Z $ we have three real roots of $Z : z_1=2, z_2=4, $ and a third real root given in terms of LambertW function: $ z_3=-\frac{2 W\left(\frac{\log ...
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1answer
36 views

Prove $f$ analytic on $D(z_0;R)\setminus\{z_0\}$ implies $\exists M, f(D(z_0;r)\setminus\{z_0\})\supset\{z\in\mathbb{C}:|z|>M\}$

Suppose $f$ is analytic on $D(z_0;R)\setminus\{z_0\}$, and $z_0$ is a pole of $f$. Prove that for any $r\in(0,R)$, there is $M\in(0,\infty)$ such that ...
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1answer
50 views

A mix of complex analysis questions.

I have a mix of complex analysis questions I can't split into other questions because they are quite trivial. In this discussion of Runge's theorem, it says: Polynomials are holomorphic, and ...
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1answer
30 views

Modular group maps upper half to itself in complex plane

Let $U$ is upper half complex plane: Suppose $$H=\{{{az+b\over cz+d}:a,b,c,d \in \Bbb R, ad-bc \gt0}\} $$ be set of modular group. Now I have to prove $H=Aut(U)$ I have some ideas, I was trying to ...
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2answers
31 views

Prove $f|_{U_0}$ is $m$-to-$1$ except at $z_0$.

Let $f$ be analytic on a domain $U$, $z_0\in U$, and $w_0=f(z_0)$. Suppose that $\mbox{ord}_{z_0}(f-w_0)=m\in\mathbb N$. Prove that there is an open set $U_0$ with $z_0\in U_0\subset U$ such that ...
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0answers
22 views

Upper bound for the ratio of Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex and $z$ is a positive real number. Do you know any results about it? Thank ...
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1answer
45 views

Is $cos(\bar z)$ analytic anywhere on complex plane?

Is $\cos(\bar z)$ analytic anywhere on complex plane ? This is my solution $d(\cos(\bar z))/dz = d(\cos(\bar z))/d(\bar z) \times d(\bar z)/dz = -\sin(\bar z) \times \lim_{\Delta z\to 0} (\Delta ...
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2answers
36 views

How to say when the integral converge and when diverge?

I have the following integral. $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. If we start solving the integral we will come up the ...
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1answer
45 views

Using Antiderivative to compute complex integral

I have been practicing solving complex integrals. When am I allowed to simply use the antiderivative of the integral to apply the Fundamental Theorem of Calculus to solve the integral? Is it when it ...
3
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1answer
48 views

Any idea how to evaluate this equation?

I'm trying to evaluate(approximate) the following integral $$ F(x,t;q) = \int_{-\infty}^{\infty}\frac{q}{q+2ik} e^{i(kx +8k^3 t)}\; dk $$ It's similar to the Airy function but I can't get rid of ...
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1answer
30 views

Partial Fractions and Complex Integral

I have $\int_{C}\frac{e^z}{z^2 + a^2}$ where $a>0$ and $C$ is a positively oriented simple closed contour containing the circle $|z|=1$. I start with $$\frac{1}{z^2 + a^2} = ...
1
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1answer
40 views

does an analytic function have a bounded derivative?

Let $G$ be a domain and $f : G \to G$ be any holomorphic function that satisfies $f(z_0) = w_0$ for $z_0, w_0 \in G$. Must it be the case that $|f'(z_0)|$ is uniformly bounded? When $G$ is the open ...
0
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0answers
30 views

find laurent series of $\frac{1}{1 - cos z} $ [duplicate]

So it is not to solve everything with regards to the series. I was talking to my professor today and he mentioned something about since my denominator is an analytic function itself I could bring it ...
1
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4answers
56 views

Find all the complex numbers $z$ satisfying

Find all the complex numbers $z$ satisfying $$ \bigg|\frac{1+z}{1-z}\bigg|=1 $$ So far I´ve done this: $$ z=a+bi \\ \bigg|\frac{(1+a)+bi}{(1-a)-bi}\bigg|=1 \\ ...
0
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0answers
35 views

a linear transformation with an inequality

Let $f$ be an analytic function mapping the open upper half plane, $\{z: \text {Im}z>0\}$ to the open unit disk, $\{z:|z|<1\}$, with $f(i)=0$. Prove that $|f(z)|\leq | \, \frac {z-i}{z+i} \,\,| ...
1
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2answers
65 views

How to prove that an analytic function must be zero on the unit disk if zero on an arc of the unit circle? [duplicate]

I have the following question:let $f(z)$ be continuous on the closed unit disk, $\{z: |z|\leq 1\}$, and analytic on the open unit disk, $\{z: |z|<1 \}$, with $f(e^{it})=0$ for $0\leq t \leq \pi/4$. ...