Tagged Questions
0
votes
0answers
36 views
How to solve for $z$ in $\dfrac{xy}{1-x}=(1-z)(x-x^{1/z})$
How to solve the following for $z$:
$$\frac{xy}{1-x}=(1-z)(x-x^{1/z})$$
where $0 < x < 1$, $\;0 < y < 1$, $\;0 < z \leq 1$.
3
votes
2answers
47 views
On the Hurwitz Zeta Function
In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
0
votes
1answer
35 views
Realization of Bessel functions
As I know when whe trying to execute Bessel functions when $z < 8$ we using this formula
$$J_\nu(z) = (\frac{1}{2}z)^\nu \sum_{k = 0}^{\infty} \frac{(-\frac{1}{4}z^2)^k}{k!\Gamma(\nu+k+1)}$$
And ...
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 ...
3
votes
2answers
186 views
Calculate an integral involving Hermite polynomials
I have to calculate the integral
$$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$
where $H_n(x)$ is the $n^{th}$ Hermite ...
0
votes
1answer
80 views
How to solve $(m_{(t)} x')' + kx = 0$ Sturm Liouville equation with bessel functions
I have been working on this problem for a while now and think I need assistance. I am trying to solve with respect to $x_{(t)}$ over the interval $t = [0, \infty]$:
$$(m_{(t)} x')' + kx = 0$$
...
1
vote
2answers
164 views
Electric Potential of an off axis charge (Legendre Generating Function)
An insulated disk, uniform surface charge density $\sigma$, of radius R is laid on the xy plane. Deduce the electric potential $V(z)$ along the z-axis. Next ...
2
votes
1answer
218 views
Normalization of the Bessel function
I would greatly appreciate assistance with the following problem.
show:
$$\int _0 ^\infty J_n(x)dx = 1; \forall n \in \mathbb{N}^+$$
for $J_o,$ use $$\mathscr{L}{J_o(at)} = \int _0 ^\infty ...
-1
votes
1answer
152 views
Integral of the square of the Bessel Function
Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object.
I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to ...
2
votes
1answer
74 views
Show that the series representation of the Bessel function works
For the following series representation of the Bessel function:
$$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$
I want to show that w is indeed the Bessel function, such ...
0
votes
0answers
45 views
if system is localized
Consider function $g=e^{-|x|}, x \in R$.
Let $\psi_n$ be a Hermite function (see http://en.wikipedia.org/wiki/Hermite_polynomials for definition). Consider system $\Psi=\{\psi_n\}, n \in N$.
Let ...
3
votes
1answer
321 views
Relationship between Legendre polynomials and Legendre functions of the second kind
I'm taking an ODE course at the moment, and my instructor gave us the following problem:
Derive the following formula for Legendre functions $Q_n(x)$ of the second kind:
$$Q_n(x) = P_n(x) ...
0
votes
1answer
61 views
Evaluating $\int_0^1 \! C(x) \, \mathrm dx$ through integration by parts
$$
\int_0^1 \! C(x) \, \mathrm{d} x.
$$
where $C(x) = \int_0^x \cos(t^2) \, \mathrm{d} t$.
I am really not quite sure how to go about this one, especially given that it needs to be calculated ...
0
votes
1answer
81 views
Finding a general coefficient in the multiplication of the two series
Help me please to find a general coefficient $a_j$ of the following series
$$
...
1
vote
0answers
80 views
integral with bessel function represented as a series [duplicate]
Possible Duplicate:
prove equality with integral and series
This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
5
votes
1answer
179 views
prove equality with integral and series
I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true
$$
...
0
votes
1answer
77 views
Condition for frame of $L_2$
Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions
$$
f_{m,k}=f^m(x-2^k)
$$
Under which conditions $\{f_{m,k}\}$ would be a ...
4
votes
1answer
87 views
Beta integral transformation
It's a homework task and I can't get past the last step.
Task is to prove that
$$
B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau
$$
By substituting ...
2
votes
0answers
46 views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
2
votes
0answers
118 views
Hermite functions and integral
Let
$$
h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2},
$$
where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function.
Consider
$$
...
1
vote
0answers
54 views
$n$-th derivative of the prolate spheroidal function
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
0
votes
1answer
57 views
Expansions of Hermite functions
I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients.
Thank you for your ...
1
vote
0answers
132 views
integral with Bessel function
Let $n$ be half an odd integer, say $n=k+1/2, k \in Z$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral
$$
\int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot ...
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 ...
0
votes
0answers
78 views
question on the expansion of the function
For a given real number $c>0$ define functions $\left(\psi_k^c(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
1
vote
1answer
326 views
An integral of a complementary error function
I really appreciate it if someone help me solving this integral:
$$ \int \frac 1x \cdot \operatorname{Erfc}^n x\, dx,$$
where $\operatorname{Erfc}$ is the complementary error function, defined as ...
1
vote
3answers
98 views
finding bound for the integral
I am trying to get bound for the following integral
$$
\int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty
$$
In particular, the bound of the form $\frac{constant}{r}$.
Sorry, we can ...
0
votes
0answers
136 views
integral of Bessel function
How to compute the following integral of Bessel function $J_1$
$$
\int_0^{\infty}|x^{-s+1}J_1^s|dx, s\geq 2
$$
Can it be computed the same way as in An integral about Bessel function
Or there are ...
4
votes
1answer
212 views
How can I solve this integral equation in terms of Hermite polynomials?
It must be proven that the solution of the integral equation
$$f(x)=\int_{-\infty}^{+\infty} e^{-(x-t)^2} g(t)dt$$
is
$$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$
...
5
votes
1answer
542 views
Integral Representations of Hermite Polynomial?
One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
6
votes
1answer
826 views
Some questions about the gamma function
Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral.
Show $\Gamma'(y) = ...
