Tagged Questions
-1
votes
1answer
72 views
Assuming the power series for $f(x)=e^x$ holds for complex numbers, show that $e^{ix}=\cos x+i\sin x$
assuming the power series for $f(x)=e^x$ holds for complex numbers, show that $e^{ix}=\cos x+i\sin x$
Question.
How can I solve this problem?
What is the complex number and what does it mean in ...
1
vote
0answers
44 views
Infinite products (involving complex numbers)
I am learning the Gamma function, based on some lecture notes, and I wish to ask a couple of questions regarding infinite products.
Let $z$ be a complex number except $\{0, -1, \ldots \}$.
(1) How ...
6
votes
1answer
84 views
Evaluating $ \sum\limits_{n = 1}^{\infty}{\binom{2n}{n} \frac{1}{5^n}} $
Here is the question a little bigger:
$$ \sum_{n = 1}^{\infty}{\binom{2n}{n} \frac{1}{5^n}} $$
This is given as an example (not an exercise), given in my textbook, on how to solve sums involving ...
1
vote
0answers
26 views
Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.
Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
1
vote
1answer
63 views
Check my work: $\lim a_n = 0 \Rightarrow \lim \sqrt{a_n} = 0 $? (for $a_n$ positive)
I'm trying to prove, as "properly" as possible the following:$$\left[ \lim z_n = z \right] \iff \left[ \lim x_n = x \quad \wedge \quad \lim y_n = y \right]$$
where $z_n = x_n + i y_n$ and $z=x+iy$.
...
1
vote
0answers
30 views
Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.
Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$
where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
-1
votes
0answers
55 views
Is the radius of convergence of the power series for $\dfrac{1}{1-z}$ centered at $0$ equal to $1$?
The radius of convergence of the power series of the function $f(z)=\dfrac{1}{1-z}$ with center $z_0=0$ is equal to $1$. Is this true or false?
1
vote
1answer
49 views
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
1.$\sum f_n$ converges uniformly on $D$?
2.$f_n$ and $f'_n$ converges uniformly on $D$?
3.$\sum f'_n$ converges on $D$ pointwise?
4.$f_n''(z)$ ...
2
votes
0answers
37 views
show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]
I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
4
votes
1answer
48 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
2
votes
1answer
50 views
Infinite Series Problem Using Residues [duplicate]
Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
...
7
votes
2answers
94 views
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
the non-zero zeros of ...
9
votes
3answers
216 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 ...
2
votes
2answers
58 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, ...
0
votes
1answer
35 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 $\partial D$.
5
votes
2answers
77 views
Determine the character of $\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$
Determine the character of the following series:
$$\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$$
where $\theta$ is a real parameter.
I try to divide the series with De Moivre' s formula:
...
2
votes
2answers
63 views
Evaluate $\sum_{n=0}^\infty \frac{\sin ((2n+1)\theta)}{(2n+1)}$
How to evaluate the value of
$$\sum_{n=0}^\infty \frac{\sin ((2n+1)\theta)}{(2n+1)}$$
My try:: I tried similar manipulating Proving a trig infinite sum using integration , but I am getting constant.
...
11
votes
2answers
200 views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
1
vote
0answers
46 views
Using Montel's Theorem to show locally uniform convergence of analytic functions
If $f_n :U \to \mathbb{C}$ is a sequence of analytic functions, where $U$ is open and connected. And there exists a point $z_0 \in U$ such that for all $k \geq 0$ the sequence $f_n^{(k)}(z)$ ...
2
votes
1answer
46 views
Showing a sequence of analytic functions converges locally uniformly
Let $f_n :U \to \mathbb{C}$ be a sequence of analytic functions on an open and connected set $U$. Suppose that the sequence is locally bounded and that for the set $$D:= \{z \in U : f_n(z) \, \, ...
2
votes
1answer
52 views
calculate the roots of $z = 1 + z^{1/2}$ using Lagrange expansion
I am trying to find the roots of equation $z = 1 + \sqrt{z}$ using Lagrange's expansion given on this book page no 15.
I expanded $z = 1 + \zeta z^p$ as
$$z = 1 + \zeta + \frac{2p}{2!}\zeta^2 + ...
1
vote
2answers
39 views
Giving a Laurent series expansion - is it alright to have two $\sum$'s in the expansion?
How would you develop the function $$f(z) = \frac{1}{z(z-1)(z-2)}$$ in the annulus $A = \{z: 1<|z|<2\}$?
I have split the function into partial fractions to obtain the result $$ -\sum_0^\infty ...
0
votes
1answer
59 views
Prove that the taylor series of cos(z) and sin(z) are holomorphic
I have a question on an old exam paper given as follows:
If we define
$$\cos(z) = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ... \frac{z^{2n}}{(2n)!} + ... = \sum_{n=0}^{\infty} ...
2
votes
4answers
110 views
Radius of Convergence of power series of Complex Analysis
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
1
vote
1answer
24 views
Prove that for n~=n' sum is much smaller than the case with n=n'
Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics.
...
2
votes
2answers
86 views
Approximating an infinite sum of only odd terms by a definite integral
Consider the infinite Sum
$S=\sum\limits_{n\ \text{odd}}^{\infty}\left(\dfrac{1}{nt}\right)^2\left[1-i\left(nt\right)^2\right]^{-1}$
Is there a way to approximate this sum as a contour integral? In ...
2
votes
3answers
70 views
Uniform limit of holomorphic functions
Let $\{f_n\}$ be a sequence of holomorphic functions defined in a generic domain $D \subset \mathbb{C}$. Suppose that there exist $f$ such that $f_n \to f$ uniformly.
My question is: is it true that ...
3
votes
1answer
113 views
Is $\sum_{n=2}^\infty\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$ convergent?
The question is motivated by the following exercise:
Find an example of a sequence of complex numbers $\{a_n\}$ such that $\sum a_n$ converges but $\prod(1+a_n)$ diverges.
A necessary condition ...
2
votes
0answers
41 views
Closed form formula for a double series related to wave equation
Does anyone have a closed form formula for the double series
$$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$
This is related ...
8
votes
1answer
148 views
The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)$
Does anyone have a proof that
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)=\frac{\pi}{8}$$
2
votes
1answer
54 views
find the series $\frac{z}{\mathrm{e}^z-1}$
How can I find the series corresponding to the function$$\frac{z}{\mathrm{e}^z-1}$$ By seeing answer given in book I came to know that series is like ...
0
votes
2answers
51 views
Complex Constant and Convergent Power Series
Suppose that the function $f$ is defined by a convergent power series and suppose that
$f (z + w) = f (z) f (w)$ for all complex $z$, $w$.
(a) Prove directly from this assumption that there is a ...
0
votes
1answer
51 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://www.math.uga.edu/~pollack/infprimes-final.pdf
Basically, they end up with $$\sum_{d \geq ...
7
votes
1answer
119 views
Does a convergent power series on a closed disk always converge uniformly?
If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
1
vote
2answers
47 views
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero.
How can I able to prove the above problem without ...
0
votes
1answer
64 views
Radius of convergence of $\sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$
I want to find out the radius of the following power series of a complex variable:
$\sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$
The ration test $\lim_{n \to ...
3
votes
1answer
162 views
Discontinuities of $\sum \frac{x^{\rho}}{\rho}$
H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
9
votes
0answers
111 views
counterexample to Abel's theorem
$D=\{z\in\Bbb C:|z|<1\}$. Let $f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)$ be a power series, the radius of convergence of $f$ is $1$, $\sum\limits_{n=0}^\infty a_n =s$.
Give a ...
0
votes
1answer
55 views
Can a power series always be integrated term-by-term inside the circle of convergence of its sum function?
Is it true that a power series can always be integrated term-by-term inside (i.e. in the interior of) the circle of convergence of its sum/limit function?
My complex analysis textbook merely states ...
5
votes
2answers
121 views
What is the order of this pole?
$$f(z)=\frac 1{\cos(z^4)-1}$$
$z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does ...
0
votes
1answer
69 views
Convergence radius of power series
I am trying to solve an exercise, but i am not sure that the result i get at the end is correct...May i kindly ask you for a little help or a remark?
Find the radius of convergence of the following ...
5
votes
1answer
74 views
Does a bounded convergent power series on an open disc extend to the boundary?
Here is my question: Suppose that $|\sum_{n=0}^{\infty}a_nz^n| \leq M$ for all $z \in D_r$ (the open disc or radius $r$). Does this power series converge on $\partial D_r$?
3
votes
1answer
148 views
How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$
Today I have encounter a series:
$$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$
where $u \not \in \Bbb{Z}$
. I have known a method to computer it (by Residue formula):
...
0
votes
1answer
24 views
How to express this inverse complex series
i know about geometric series in the normal form, but i've been incapable of find an expression for this series, any suggestions
$$\sum_{n=-\infty}^{0}(2e^{-jw})^{n}$$
Thanks
2
votes
0answers
84 views
Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$
Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...
1
vote
1answer
52 views
Convergence of a sum of rational functions
I have the complex functions $f_n(z) = 1/(1+z^n)$ and I'm supposed determine where $\sum_{n=1}^\infty f_n(z)$ converges for $z \in\mathbb{C}$
Extra Info:
I was only able to determine convergence ...
5
votes
4answers
243 views
How do I calculate $\sum_{n\geq1}\frac{1}{n^4+1}$?
How do I calculate the following sum $$\sum_{n\geq1}\frac{1}{n^4+1}$$
1
vote
0answers
52 views
what are the borders of the convergence disks of series?
Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series:
$$\sum_{k=1}^\infty{a_kz^k}$$
is convergent on ...
3
votes
0answers
141 views
Under which hypotheses is switching between sum and integral signs legit?
Which hypotheses are needed to change the order of sum and integral signs?
Concrete example: consider the expression
$$
...
1
vote
4answers
117 views
complex power series
Sorry for taking this from another question, but the second part was never answered, and I'm not sure how to get there. From: Prove the following equation of complex power series.
Show that for $|z| ...
