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, ...
0
votes
0answers
9 views
Karman - Trefftz transform.
Is the Joukowsky transformation parameter a = b - abs(s)
Where s is the coordinates of the center of the circle with radius b, the same as the one for Karman-Trefftz transform for a slightly offset ...
2
votes
1answer
51 views
Is there a geometric relationship between plane geometry and polynomials?
It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a ...
4
votes
4answers
101 views
what is the relation of smooth compact supported funtions and real analytic function?
What is the major difference between real analytic and test function (smooth compact supported functions). Can we find a real analytic function $f$ on $R^n$ which is also smooth compact supported? If ...
2
votes
1answer
84 views
$\int_{-\infty}^{\infty}\!e^{- \pi (x+iy)^2}\,dx = 1$ for all $y$.
Can anyone provide a proof of why $\int _{-\infty} ^ {\infty} e^{-\pi (x+iy)^2} dx$ equals 1, for all y ? $x$ and $y$ are real numbers.
EDIT: We already know this for y=0.
Thank you.
5
votes
4answers
138 views
Are Taylor series and power series the same “thing”?
I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
5
votes
3answers
76 views
A boundary version of Cauchy's theorem
I am looking for a reference for the following theorem (or something like it) that is not Kodaira's book.
Let $D$ be a domain and $\overline{D}$ be it's closure. Suppose that $f:\overline{D} ...
0
votes
1answer
22 views
Differentiable on open unit disc
I have come across the following questions on a past exam paper, but I'm not sure how to go about answering the,. Any help would be greatly appreciated....
(i) Does there exist a function $f$ ...
8
votes
3answers
184 views
Evaluating $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform
Inspired by this post, I'm trying to evaluate $\displaystyle \sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$ using the inverse Mellin transform. But my answer is twice ...
3
votes
4answers
104 views
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$?
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is ...
2
votes
0answers
39 views
Interchanging the limiting operations
How to remember the conditions for interchanging the limiting operations , for example between limits and integrals or integrals and sums or derivation of any order and integrals, i mean every one of ...
1
vote
1answer
73 views
question about complex analysis log function
I have this domain:
$$D = \{z \in\Bbb C: \text{Re}(z) \leq \text{Im}(z) < \text{Re}(z) + 2\pi\}.$$
We know that the exponential function restricted to D is one-to-one
The log is defined as: Log: ...
4
votes
2answers
155 views
Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle ...
2
votes
1answer
33 views
Evaluating Complex Line Integrals
Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented.
This is what I have tried to do, starting ...
2
votes
1answer
41 views
Evaluating $\sum_{n=0}^\infty \frac 1 {(2n+1)^4}$ using Mittag-Leffler's expansion
I am trying to evaluate the following series using Mittag-Leffler's expansion theorem. What function would be useful?
$$\sum_{n=0}^\infty \frac 1 {(2n+1)^4} = \frac{\pi^4}{96}$$
I considered ...
0
votes
1answer
64 views
Evaluating the improper integral $ \int_{0}^{\infty}{\frac{x^2}{x^{10} + 1}\mathrm dx} $ [duplicate]
I am trying to solve the following integral, but I don't have a solution, and the answer I am getting doesn't seem correct.
So I am trying to integrate this:
$$ \int_{0}^{\infty}{\frac{x^2}{x^{10} ...
1
vote
1answer
36 views
Parseval's identity
How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
0
votes
1answer
45 views
$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, ...
0
votes
0answers
43 views
Complex Analyisis: Exponential Function
Suppose I have this domain:
$$D = \{z \in\Bbb C: \text{Re}(z) \leq \text{Im}(z) < \text{Re}(z) + 2\pi\}.$$
I have to show that the exponential function restricted to $D$ is one-to-one.
To prove ...
3
votes
1answer
47 views
Do there exist some non-constant holomorphic functions such that the sum of the modulus of them is a constant [duplicate]
Do there exist some non-constant holomorphic functions $f_1,f_2,\ldots,f_n$such that $$\sum_{k=1}^{n}\left|\,f_k\right|$$ is a constant? Can you give an example? Thanks very much
1
vote
0answers
49 views
Why is this function continuous on $ x \in \mathbb{R} $ where $ x < 1 $? [duplicate]
The function is
$$ \exp(\frac{1}{3}\text{log}[(z-1)(z-2)(z-3)]) $$
where
$$ \log[(z-1)(z-2)(z-3)] = \int_4^z{\frac{[(z-1)(z-2)(z-3)]'}{[(z-1)(z-2)(z-3)]}dz} + \log 6 $$
This function is clearly ...
0
votes
0answers
19 views
$x,y\in\mathbb{C}^n$, $f(x,y)=\sup_{\theta,\phi}\{||e^{i\theta}x+e^{i\phi}y||^2,\theta,\phi\in\mathbb{R}\}$
$x,y\in\mathbb{C}^n$, $f(x,y)=\sup_{\theta,\phi}\{||e^{i\theta}x+e^{i\phi}y||^2,\theta,\phi\in\mathbb{R}\}$ Then which is/are the following are true?
$1. f(x,y)\le ||x||^2+||y||^2+2|<x,y>|$
...
0
votes
1answer
33 views
Using complex logarithms to solve equations
Could someone please just explain the formula/method for solving the complex equation $$e^{iω}=k$$ where $k∈C$.
As an example, I know that when $ω=x+iy$, $e^{2iω}=1$ has solutions $ω = n\pi$ for ...
1
vote
2answers
48 views
Complex integral help involving $\sin^{2n}(x)$
Show that $$\int_0^\pi \sin^{2n} \theta d\theta=\dfrac{\pi(2n)!}{(2^n n!)^2} $$
So far I have came up with:
$$\sin^{2n} \theta = \left(\dfrac {z-z^{-1}}{2i} \right)^{2n}$$ and I know I should be ...
1
vote
0answers
48 views
Approximating the modified Bessel’s function with a sum of exponentials
I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
3
votes
1answer
89 views
Inverse Laplace transform of the function: $F(s)=e^{-a\sqrt{s(s+r)}}$
I would like to find inverse Laplace transform of the function: $$F(s)=e^{-a\sqrt{s(s+b)}}$$
which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be appreciated if ...
4
votes
2answers
44 views
Radius of convergence of the Bernoulli polynomial generating function power series.
The generating function of the Bernoulli Polynomials is: $$\frac{te^{xt}}{e^t-1}=\sum_{k=0}^\infty B_k(x)\frac{t^k}{k!}.$$
Would it be right to say that the radius of convergence of this power series ...
1
vote
3answers
45 views
Laurent series for $\frac{1}{(z-1)(z+1)}$ centered at $z = 1$
practice problem got us stuck -- please help us!!! Thank you..<3
Find a Laurent series for $\frac{1}{(z-1)(z+1)}$ centered at $z=1$ and specify the region in which it converges.
What we did was ...
1
vote
1answer
37 views
What is the radius of convergence of a power series in two variables?
What is the radius of convergence of a power series in two real variables?
If I were to fix one of the variables (i.e. make it a real constant), then would the radius of convergence simply be related ...
4
votes
2answers
77 views
singularity of analytic continuation of $f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2}$
How to show that all possible collection of analytic continuations of $\displaystyle f(z) = \sum_{n=1}^\infty \frac{z^n}{n^2} $ has singular point at $z = 1$. I know that $f(z)$ converges for $|z| \le ...
1
vote
1answer
63 views
To show that $\sqrt{z^2-1}=\exp(\frac{1}{2} \log(z^2-1))$ is analytic in the plane minus [-1,1]
By the definition of logarithm branch, we set that $\sqrt{z^2-1}=\exp(\frac{1}{2} \log(z^2-1))$
However to show that the $\sqrt{z^2-1}$ is analytic in the entire plane minus the interval [-1,1], it ...
2
votes
2answers
53 views
If $|f(z)| \leq |z|^2+\frac{1}{\sqrt{|z|}}$, show f is quadratic polynomial.
Suppose the function f is analytic in the punctured plane $z!=0$ (it means we excluded the zero) and satisfies the above condition, $|f(z)| \leq |z|^2+\frac{1}{\sqrt{|z|}}$, then show f is quadratic ...
3
votes
4answers
77 views
Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?
How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
2
votes
0answers
29 views
Show that the given function is analytic in the entire plane minus the interval [1, 2].
This problem is for the complex analysis.
The problem is followed by this, show that ${[(z-1)(z-2)^2]}^{1/3}=e^{\frac{1}{3}f(z)}$ is analytic in the entire plane minus the interval [1,2].
It seems ...
1
vote
1answer
18 views
$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when
$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when,
$1.|z|\le 3$
$2. |z|<\sqrt{3}$
$3.|z-1|<\sqrt{3}$
$4.|z-1|\le \sqrt{3}$
The radius of convergence can be found by applying the root ...
0
votes
1answer
33 views
Maximum Modulus theorem applied on mapping
Question: For $|z_0|<R$, I want to show that the mapping $$T(z)=\frac {R(z-z_0)} {R^2-\bar{z_0}z}$$ takes the open disc of radius $R$ $1-1$ and onto the unit disc and $z_0\rightarrow 0$.
Hint: ...
1
vote
2answers
61 views
Making a cube root function analytic on $\mathbb{C}\backslash [1,3]$
I am still not convinced by the post that the function$$\sqrt[3]{(z-1)(z-2)(z-3)}$$ can be defined so it is analytic on $\mathbb{C}\backslash [1,3]$. We define for each $z\in \mathbb{C}\backslash ...
1
vote
1answer
47 views
Contour Integral and Complex Identity
While studying I came across this problem:
(a) For $z=x+iy$, show that
$$|\cos \pi z|^2=\frac{1}{2}(\cos(2\pi x)+\cosh(2\pi y))$$
(b) For a positive integer let $\gamma N$ be the square connecting ...
9
votes
0answers
92 views
Is there a classification of isolated essential singularities?
In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked:
Do we not care about essential singularities to classify ...
2
votes
2answers
81 views
Recursion relation for Euler numbers
I am trying to solve the following:
The Euler numbers $E_n$ are defined by the power series expansion
$$\frac{1}{\cos z}=\sum_{n=0}^\infty \frac{E_n}{n!}z^n\text{ for }|z|<\pi/2$$
(a) ...
3
votes
2answers
48 views
Laurent Series Expansion
Give the Laurent series development of the function $f(z)=\frac{1}{z(z-1)(z-2)}$ in the three rings $A_1=\{z:|z|<1\}$, $A_2=\{z:1<|z|<2\}$ and $A_3=\{z:2<|z|\}$.
I have gotten the partial ...
1
vote
1answer
50 views
is this solution of $\int_0^\infty \sin(z^2) dz $ valid?
Is this method valid?
We have $\displaystyle \int_0^\infty e^{-u^2}du = \frac{\sqrt \pi}{2}$.
Let $u = \frac{1+i}{\sqrt 2}x$, we get $\displaystyle \frac{1 + i}{\sqrt 2 }\int_0^{\infty }e^{-i x^2} ...
0
votes
2answers
30 views
contradicting identity theorem?
the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$
Let $f(z) = \sin ...
2
votes
1answer
34 views
Question on a French math text by P. Masani
I'm trying to translate a paper on matrix-valued functions by P. Masani, published in C. R. Acad. Sci. Paris, 249, 1959, p. 873. (see http://temp-share.com/show/FHKd40yi6 for scanned pdf). My specific ...
4
votes
1answer
138 views
Integrate: $\int_0^\infty \frac{\sin^2(x)}{x^2}dx$
I am trying to integrate $\displaystyle \int_0^\infty \frac{\sin^2(x)}{x^2} dx$ by method of contour. I am considering the following contour but I am not being able to. Also I am not sure if it's ...
2
votes
1answer
29 views
Contour Integral
I have this question:
I'm aware that $e^{iz^2}$ is analytic, and hence $I_R = 0$ by Cauchy's Integral theorem. I'm not really sure what to do from there. Thanks!
1
vote
1answer
22 views
Reciprocal of Laurent series
I understand how to use long division to find the reciprocal of a power series.
Are there any theorems that allow you to divide Laurent series that have a finite number of terms with negative ...
2
votes
2answers
56 views
Holomorphic series with its real part positive $f(z)=1+\sum_{n=1}^\infty a_n z^n$
Let $$f(z)=1+\sum_{n=1}^\infty a_n z^n,$$
$f \in H(B(0,1))$, and $\operatorname{Re} f(z)\ge 0$,
$\forall z \in B(0,1) $.
Prove:
(1) $| a_n | \le2$;
(2) $|a_1^2-a_2| \le 2, ...
1
vote
1answer
44 views
Composition of $\mathrm H^p$ function with Möbius transform
Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$
Consider a Möbius ...
0
votes
1answer
33 views
complex holomorphic function which only has finite roots
Suppose that D is a bounded region, f $\in$ H(D)$\bigcap$C($\bar D$).Prove that f has only finite roots if f$\neq$0 on $\partial D$.
6
votes
2answers
50 views
a problem on nonconstant holomorphic function has a zero or no in the closed unit disk
Let $f:D \to \mathbb{C}$ be a non-constant holomorphic function ($D$ is the closed unit disk) such that $|f(z)|=1$ for all $z$ satisfying $|z|=1$ . Then prove that there exist $z_0 \in D$ such that ...
