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
24 views

laurent series expansion problem 1

I am trying to find the laurent series for the function $\frac{1}{z+z^2}$ for domain $0<|z+1|<1$. I separated the function into: $f(z)=\frac{1}{z}\frac{1}{z+1}$ I am having trouble with the ...
0
votes
0answers
22 views

Reference Request for Complex Analysis (with some specificity regarding Ahlfors and Cartan)

I am a self-studier and am making my second pass through Complex Analysis. I have read the reference request posts many times. Yet perhaps I could get some advice as to the relevant benefits of ...
2
votes
1answer
38 views

Simple Residue calculation

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$ Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...
2
votes
3answers
47 views

Can a non-constant analytic function have infinitely many zeros on a closed disk?

I think not, however my proof is quite sketchy so far.. My attempt: Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there exists a sequence of zeros in D with a ...
1
vote
0answers
32 views

The inverse of the Gamma function at $-\infty$

Let $\Gamma$ be the analytic continuation of the Gamma function $$\Gamma:z\mapsto \int_0^{+\infty} x^{z-1}e^{-x}dx$$ on the complex plane except non-positive integers. We know that $\Gamma$ has no ...
1
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0answers
36 views

Complex Exponential in Differential Equations.

I am a physics student, but have taken courses in analysis and algebra. My knowledge of differential equations is purely methodical, and I was hoping for a more math oriented insight with regards to ...
-1
votes
1answer
36 views

Show that $g(z) =\bar{z}$ is continuous at every point of $\mathbb{C}$ and that it's not differentiable at any point. [closed]

Show that $g(z) =\bar{z}$ is continuous at every point of $\mathbb{C}$ and that it's not differentiable at any point.
0
votes
0answers
27 views

What is An Image (of the Riemann Sphere)? [closed]

What's the image of the left half plane of the Riemann Sphere? {z in C; Re(z)<0} How do I find the image??
0
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1answer
22 views

Taylor Series Expansion for ${z^2+4z^4+z^6}/(1-z^2)^3$

So I know for sure that the Taylor Series expansion for $1/(1-z^2)^3$ is $\sum {k(k-1)z^{2k-4}/{2}} $ assuming |x|<1. But what do we do with the top? I think its already in the expanded form, ...
-5
votes
2answers
43 views

Integrate x^(2m)/(1 + x^(2n)) from 0 to infinity, where 0 <= m < n. [closed]

Integrate $x^{2m}/(1 + x^{2n})$ from $0$ to $\infty$, where $0 \le m < n$.
2
votes
1answer
76 views

Rearranging $\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k$.

Consider the complex series: $$\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k.$$ Clearly the series converges for $\left|z+\frac{1}{2}\right| < 1$, by the ratio test. I am supposed to write ...
1
vote
0answers
15 views

Map in $C(\mathbb{T})$ with constant modulus 1

I know that a function in A($\mathbb{D})$ (analytic on the open disk and continuous on its boundary $\mathbb{T}$) with constant modulus on $\mathbb{T}$ is a finite Blaschke product. But what about a ...
0
votes
1answer
23 views

How to compute the following real integrals using the residue theorem?

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
6
votes
1answer
72 views

Holomorphic function injective on annulus => injective on unit disk?

Let $f(z)$ be analytic on the unit disc, and suppose that there is an annulus $U =$ {$z ∈ C| r < |z| < 1$} such that $f(z)$ restricted to the annulus U is injective. Show that f is injective on ...
0
votes
1answer
22 views

Taylor Series expansion and radius of convergence for $e^z+e^{-z}+2cosz/4$

So I did this by taking apart bits of that long equation: $e^z=\sum z^n/n!$, $e^{-z}=\sum(-z)^n/n!$ $2\cos z=e^{iz}+e^{-iz}$ So when we put these together as a Taylor Series, do we just add them ...
1
vote
1answer
23 views

How does squaring a function affect it's removable singularities?

This is a simple question. say you have a function, f, with a removable singularity. does f^2 have a removable singularity at the same point? I strongly suspect that squaring the function would only ...
3
votes
1answer
102 views

Let f be analytic on ∆

The problem is: let $f$ be an analytic function on $\Delta$ and satisfy $|f|<1$. Prove that if $f(1/2)=f(−1/2)=0$, then $|f'(0)|\le 1/4$. I tried to expand $f$ at $0$ and then plug in $1/2$ and ...
5
votes
0answers
32 views

Affine curve is union of $d$ lines through point of multiplicity $d$. [closed]

Let $C$ be an affine curve defined by a polynomial of $P(x, y)$ of degree $d$. Show that if $(a, b)$ is a point of multiplicity $d$ in $C$ then $P(x, y)$ is a product of $d$ linear factors, so $C$ is ...
3
votes
1answer
296 views

How to properly translate the coefficients of a Taylor series?

Given a Taylor series $$f(z) = \sum_{k=0}^\infty c_k^{(a)}\frac{(z-a)^k}{k!}$$ of a meromorphic function $f$ in $\mathbb C$ (i.e. analytical except for a set of isolated points) around some value ...
1
vote
1answer
53 views

Residue theorem

Let us say we need to perform the classic integral $$ I=\int_{-\infty}^{+\infty}dz \,\frac{e^{itz}}{z^2+1}~, $$ where $t>0$. What is normally done is the following. We consider the integral $$ ...
2
votes
1answer
9 views

How to find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them

Find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them. I'm fairly sure that due to the exponential term overpowering the factor $z$, there will be an essential singularity ...
1
vote
2answers
35 views

When the argument of complex numbers is a well defined real valued function?

I know that the argument $\arg:\Bbb C\setminus\{0\}\to\Bbb R$ is multivalued function and also that if we consider $\arg:\Bbb C\setminus\{0\}\to{\Bbb R}/{2\pi \Bbb Z}$, then it is a well defined ...
0
votes
1answer
17 views

Find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$

I have to find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$ for $|a| <R < |b|$ I would use Cauchy formula but first what can I do with $\frac{1}{(z-b)(z-a)^m}$? I dont remember it.
2
votes
1answer
429 views

complex main branch of a logarithmic function holomorphic correct

Let $z:= x+iy$ It will now be shown that $$f(z)= f(x+iy) = \frac{1}{2} \log(x^2 + y^2) + i \arctan\left(\frac{y}{x}\right); (z\in \mathbb{C}, x = \operatorname{Re} z \ne 0),$$ where $arctan$ ...
1
vote
2answers
72 views

Can the hypergeometric function be extended analytically to the complex plane in the interval [1,$\infty$ )?

Just a thought. The hypergeometric function, which can be written as: $$F(a,b,c \space;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$$ is obviously ...
4
votes
2answers
347 views

If |f| is constant, f is constant.

I am confused as to how they got from the two equations being equal to 0 to the derivative being 0. I could be really tired right now but this isn't really making sense to me. I was thinking of ...
0
votes
1answer
30 views

Distance on riemann sphere [duplicate]

Let we have $C$ the set of complex numbers and $z_1 , z_2 \in C $ we have $Z_1 , Z_2 \in S$ correspond on riemann sphere and we will define : $$ d(Z_1,Z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2} ...
6
votes
0answers
121 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
1
vote
2answers
24 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
0
votes
0answers
15 views

Where exactly is coupling used in this probablistic proof of Liouvelle's theorem? [closed]

In the last section of the following blog post https://blameitontheanalyst.wordpress.com/2012/01/24/probabilistic-coupling/ coupling is supposedly used but it is not clear at which step they ...
0
votes
0answers
31 views

Counterexample to Line Integral depending on end points

Theorem The line integral $$\int_\gamma p\, dx + q\,dy$$ defined in $\Omega$, depends only on the end points of $\gamma$ if and only if there exists a function $ U(x,y) $ in $\Omega$ with the ...
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0answers
16 views

Switching between Cartesian coordinate and polar coordinates

Under what assumption, every non-zero complex number represented in Cartesian coordinate system admits unique polar representation and vice versa ?
0
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0answers
29 views

Holomorphic functions (continuity of partial derivatives)

Let $f:\Omega\rightarrow \mathbb{C}$ be an holomorphic function i.e. for any $z_0\in \Omega$ there exists the limit: $$f^{'}(z_0) = \lim_{z\mapsto z_0}\frac{f(z)-f(z_0)}{z-z_0}.$$ Let us write $f(z) ...
2
votes
2answers
33 views

Prerequisites for Silverman's Arithmetic of Elliptic Curves

I would like to take a course on elliptic curves using Silverman's Arithmetic of Elliptic Curves next year. I would be taking complex analysis concurrently, but it was listed as a formal prerequisite, ...
0
votes
1answer
17 views

Unsure about expansion

Hello, can someone tell me how this expression is expanded in this proof. Does it follow from some other theorem?
0
votes
1answer
24 views

Show that $\sum r^n cos(nx)=rcos(x)-r^2/(1-2rcos(x)+r^2)$

I'm a little unsure about how to approach this. I've been told that we have to use the relationship that $\sum r^n=1/1-r$. However, I'm not too sure what to do with the $\cos(nx)$. Can someone give ...
1
vote
1answer
42 views

Find and show that the residues of the meromorphic differential $dx$ for Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ is zero

Find the residues of the meromorphic differential $dx$ of Fermat's curve $C:$ $\xi_1^4+\xi_2^4=\xi_0^4$ at its poles. Check that their sum is zero, as it must be. Attempt: Let $\xi_2\not=0$. Then ...
0
votes
1answer
111 views

complex analysis poles and residues

I am trying to understand a lemma on the (end of the first page - second page) on this link: http://math.uga.edu/~pollack/infprimes-final.pdf Basically, they end up with $$\sum_{d \geq 1}f(d) ...
1
vote
1answer
14 views

Constructing an entire function with a given isolated zero set

We know that a nonzero entire function on $\mathbb C$ has an isolated set of zeroes. Is it the case that, given an isolated set, there is an entire function which vanishes precisely on that set? If ...
4
votes
4answers
156 views

A question regarding Frobenius method in ODE

Suppose $b(x),c(x)$ are real functions analytic at $0$. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
2
votes
1answer
49 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 ...
0
votes
0answers
27 views

Real Cross Ratio Example

Theorem The cross ratio $(z_1, z_2, z_3, z_4) $ is real if and only if the four points lie on a circle or on a straight line. I know by geometry I can obtain $arg(z_1, z_2, z_3, z_4) = ...
2
votes
1answer
63 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 ...
1
vote
1answer
38 views

Find a complex-differentiable function with real part $x^2(ay+8) +4y^2(y+b)$

Find a complex-differentiable function $f$ with real part $u(x,y) = x^2(ay+8) +4y^2(y+b)$ I have tried to use Cauchy-Riemann to get $v(x,y)$ but realised that I need to find the constants $a$ and ...
0
votes
0answers
28 views

Is every natural recursive relation necessarily holomorphic?

Define the set of algebraic primitive recursive relations as the set of functions defined by: $$ F(n,a,k) = F(n-1,F(n-1,F(n-1,a,a),a)...,a)_{\text{nested to depth k}}$$ $$ F(0,a,k) = a + k $$ Along ...
0
votes
2answers
33 views

Analyticity of complex derivative and conjugate

Given a complex function $\phi(z)$ that is analytic, can I say that the following are analytic? 1) $\phi '(z)$ 2) $\overline{\phi '(z)}$ 3) $z\overline{\phi '(z)}$ My end goal is to compute the ...
3
votes
1answer
57 views

Therem of Residue application

I want to determinate the following integral: $$\int_{\gamma} \frac{e^z}{\cos{(z)}} dz$$ Where $\gamma (t)=\frac{\pi \cos t}{1 +\sin^2 t}(1+i\sin t)$, $0\leq t \leq 2\pi$ So I see that ...
1
vote
0answers
23 views

Application of Reflection Principle

Let $f(z)$ be an entire function whose modulus is constant on some circle. Show that $f(z)=c(z-z_0)$ for some $n\geq 0$ and some constant $c$, where $z_0$ is the center of the circle. So far I have ...
0
votes
0answers
64 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 ...
-1
votes
1answer
40 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 ...