Tagged Questions
3
votes
1answer
41 views
How to prove this identity for ${}_3F_2$ (Generalized Hypergeometric Function)?
This may look like homework, but it is not. I've found this identity (using Mathematica):
$$
{}_3F_2 \left( \matrix{1,1,1 \\ 2, e} ; 1 \right) = (e-1) \psi^{\prime}(e-1),
$$
valid for $e$ with ...
3
votes
0answers
93 views
Identity involving $\zeta(3)$
This is a follow-up of this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that
...
1
vote
0answers
42 views
Gamma Function Problem
Hi is it fair to write $$\Gamma(1+ix)=ix\Gamma(ix) $$
and then to say that $\Gamma(ix)=\exp(i\arg(\Gamma(x)))$. If not can anyone explain what $\arg(\Gamma(x))$ is defined as please, I always get ...
0
votes
0answers
12 views
Semiperiod of $\wp$
If we let $u$ and $v$ span a lattice of $\mathbb{C}$, then for the associated $\wp$-function: $\wp(\frac{u}{2})$ is algebraic over $\mathbb{Q}(G_k(u,v),k \ge 4)$ because it is a zero of $$4X^3 - ...
2
votes
0answers
39 views
Closed form formula for a double series related to wave equation
Does anyone have a closed form formula for the double series
$$\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)?$$
This is related ...
8
votes
1answer
145 views
The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)$
Does anyone have a proof that
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)=\frac{\pi}{8}$$
4
votes
1answer
117 views
Integral of $\ln |\sin(x)|$
Does anyone have a real formula for the integral $$\int\ln |\sin(x)|\,dx ?$$
Neither Maple nor Mathematica give a real answer.
Using integration by parts and the series for $x\cot x$, I get $$x\ln ...
1
vote
0answers
50 views
What is the closed form expression for this?
Let $r_1...r_k$ be the $k$ roots unity or solutions to the expression $x^k = 1$
What is the expression:
...
4
votes
1answer
119 views
Dirichlet L-series and Gamma function question
Could someone help me, please, with this exercise?
Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$.
Define the ...
4
votes
1answer
109 views
Prove that $\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}$
Does anybody know how to prove this identity?
$$\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}\quad p+q>1,q<1$$
$B(x,y)$ denotes Beta ...
0
votes
2answers
248 views
4
votes
1answer
77 views
Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$
Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$
I don't know how to do this ?
Note that $\gamma $ is the Euler-Mascheroni constant
1
vote
1answer
74 views
asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$
consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ :
$$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$
And:
$$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$
...
2
votes
1answer
70 views
About the values of the $\Gamma$ function
The $\Gamma$ function is defined by
$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$
where $z$ is a complex number.
We know that if $z$ is real then the values of $\Gamma$ are also real. I am ...
2
votes
0answers
47 views
asymptotics of $ J_{iu} (ia)$ for a Bessel function
Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.)
In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...
2
votes
1answer
70 views
What is the “name” of this function?
There is a function I met in complex analysis.
$$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
2
votes
1answer
138 views
Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$
I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...
2
votes
1answer
172 views
solution of Lagrange differential equation are square integrable
I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
6
votes
0answers
226 views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though ...
2
votes
0answers
79 views
Modified Bessel function
I use the standard notations. When $x$ is real then by definition
$$
I_{\nu}(x)=e^{-\nu\pi i/2}J_{\nu}(ix).
$$
I want to define $I_{\nu}$ for complex $z$. Watson (Treatise of the Theory of Bessel ...
4
votes
3answers
220 views
Limiting behavior of gamma function
I am trying to determine whether $\Gamma(x+iy)\rightarrow 0$ as $y\rightarrow\infty$.
How should I go about doing it?
I was trying to see if I could get anything from ...
1
vote
2answers
299 views
How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?
How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire?
Could you give me some scratch proof?
3
votes
0answers
101 views
Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$
We consider the following function.
$$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$
$u(x)$ is defined on $[-1, 1]$.
Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
5
votes
2answers
179 views
Detailed proof of $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$
I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to ...
5
votes
3answers
397 views
Derivatives of the Riemann zeta function at $s=0$
It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence :
$$\delta_{n}=\left | ...
5
votes
1answer
145 views
Topology of Branch Cuts and Elliptic Integrals
In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces.
On page 4, they discuss the integral
$$\phi(x) = ...
3
votes
2answers
341 views
Are Complex Substitutions Legal in Integration?
This question has been irritating me for awhile so I thought I'd ask here.
Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals:
...
4
votes
2answers
499 views
Euler's product formula for $\sin(\pi z)$ and the gamma function
I want to derive Euler's infinite product formula
$$\displaystyle \sin(\pi z) = \pi z \prod_{k=1}^\infty \left( 1 - \frac{z^2}{k^2} \right)$$
by using Euler's reflection equation ...
9
votes
1answer
208 views
On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others
For p = 2, we have,
$\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$
It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then,
...
3
votes
0answers
106 views
Weierstrass $\wp$-Function Addition Property
Consider the function
$$
\det\left( \begin{array}{ccccc}
&1 &\wp(z) &\wp'(z) \\
&1 &\wp(w) &\wp'(w) \\
&1 &\wp(-z-w) &\wp'(-z-w) \end{array} \right)=f(z)
$$
I'm ...
3
votes
0answers
97 views
Equivalent Definitions of the Weierstass $\wp$-Function
I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent.
Definition 1
$\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
1
vote
0answers
207 views
Using Rouche's theorem
Let $p>1$.
Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$.
Function $\phi(p)$ is analytic on its domain.
It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin ...
7
votes
1answer
315 views
Physical interpretation of the generating function for the Bessel functions.
It is well known that the generating function for the Bessel function is
$$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$
So, we have
$$f(z) = \sum_{\nu = -\infty}^{\infty} ...
8
votes
2answers
221 views
Prove that $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's identity.
I'm trying to derive the functional equation $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's formula:
...
3
votes
1answer
214 views
An addition property of Weierstrass $\wp$
I want to show
$$
\left( \begin{array}{ccccc}
&1 &\wp(v) &\wp'(v) \\
&1 &\wp(w) &\wp'(w) \\
&1 &\wp(v+w) &-\wp'(v+w) \end{array} \right)=0
$$
...
10
votes
1answer
1k views
Residue of $z^2 e^{1/\sin z}$ at $z=\pi$
A while back I was working through many problems in Mathews and Walker's Mathematical Methods of Physics. In the appendix is this problem:
A-6. Find the residue of the function $z^2 e^{1/\sin z}$ ...
2
votes
2answers
221 views
On the modulus of $\Gamma(z)$
In about two weeks, I'm going to be giving a presentation on the complex-valued Gamma function $\Gamma(z)$. By definition, I know that $$\Gamma(z)= \int_0^\infty e^{-t}t^{z-1}dt.$$
Now if I let ...
9
votes
0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
1
vote
0answers
117 views
analytic continuation of an integral involving the mittag-leffler function
i have posted this question on MO, and i didn't get an answer . we have the following integral :
$$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 ...
5
votes
2answers
189 views
Weierstrass $\wp$ function doubly periodic
I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
1
vote
1answer
284 views
Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus
I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus.
On this Wiki page, I find ...
1
vote
3answers
97 views
finding bound for the integral
I am trying to get bound for the following integral
$$
\int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty
$$
In particular, the bound of the form $\frac{constant}{r}$.
Sorry, we can ...
19
votes
1answer
452 views
Upper bound on differences of consecutive zeta zeros
The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
11
votes
3answers
485 views
Calculating $ \int _{0} ^{\infty} \frac{x^{3}}{e^{x}-1}\;dx$
how to calculate $$\int_0^\infty \frac{x^{3}}{e^{x}-1} \; dx$$
Be $q:= e^{z}-1 , p:= z^{3}$ , then $e^{z} = 1 $ if $z= 2\pi n i $, so the residue at 0 is : $$\frac{p(z_{0})}{q'(z_{0})} = 2\pi ...
2
votes
1answer
264 views
How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?
This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not.
In the question mentioned above, ...
1
vote
0answers
98 views
Bound on Bessel function of the first order
Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument.
Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?
0
votes
0answers
96 views
Concept of Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...
2
votes
2answers
351 views
Euler Gamma function $\Gamma(z)$ on $\mathbb{C}$
I'm working on an exercise about the Gamma Function from Euler.
First, $\Gamma (z)= \int_0^\infty e^{-t}t^{z-1}dt$.
Now, if we consider the "similar" function $\int_{\frac{1}{n}}^\infty ...
12
votes
3answers
293 views
How this integral $ \int_0^z\frac{1-e^x}{x} dx$ is connected to the Gamma function and Euler constant?
This is my first question in this forum; I hope it is an appropriate question.
The Wolframalpha website tells me that
$$
\int_0^z\frac{1-e^x}{x} dx = \log (-z)+\Gamma(0, -z)+\gamma\quad ...
6
votes
1answer
526 views
On the growth of the Jacobi theta function
So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5:
Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function
$$\theta(z | r) = ...
