# Search results

1. ### Hypergeometric function problem

Yes I could (\frac{1}{2})_n=\frac{(2n-1)!!}{2^n} (\frac{3}{2})_n=\frac{(2n+1)!!}{2^{n-1}} So I get _2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x)=\sum^{\infty}_{n=0}\frac{(2n-1)!!}{(2n+1)2^{n+1}n!}x^n=\sum^{\infty}_{n=0}\frac{(2n-1)!!}{2(2n+1)(2n)!!}x^n And I don't know what to do know.
2. ### Hypergeometric function problem

Homework Statement Calculate _2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x) Homework Equations _2F_1(a,b,c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n (a)_n=a(a+1)...(a+n-1) The Attempt at a Solution (\frac{1}{2})_n=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-1}{2}...
3. ### Bessel function summation

Homework Statement What is easiest way to summate \sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}] where ##i## is imaginary unit. Homework Equations The Attempt at a Solution I don't need to write explicit Bessel function so in sum could stay C_1J_(x)+C_2J_2(x)+... Well I see that...
4. ### Power series identity

Is there some other way to do it. Easier? Here I go from Taylor to Laurent series.
5. ### Power series identity

Homework Statement Show e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Homework Equations J_k(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{(n+k)!n!}(\frac{x}{2})^{2n+k} The Attempt at a Solution Power series product (\sum^{\infty}_{n=0}a_n)\cdot (\sum^{\infty}_{n=0}...
6. ### Inverse Laplace transform

\sin at*\sin at=\int^t_0\sin aq\sin (at-aq)dq=\int^t_0\sin aq(\sin at\cos aq-\sin aq\cos at)dq So we have to solve to different integrals \sin at\int^t_0 \sin aq \cos aqdq=\frac{1}{2}\sin^3 at and \cos at\int^t_0 \sin^2 aqdq=\cos at\int^t_0\frac{1-\cos 2aq}{2}dq=\frac{1}{2}t\cos...
7. ### Inverse Laplace transform

Homework Statement Find inverse Laplace transform \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}] Homework Equations The Attempt at a Solution I try with theorem \mathcal{L}[f(t)*g(t)]=F(s)G(s) So this is some multiple of \mathcal{L}[\sin at*\sin at] So \mathcal...
8. ### Poisson kernel

Is ##r## always positive? Is ##P(-r,x)## defined?

Homework Statement Pade approximation [N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D} With this approximation we approximate Maclaurin series f(x)=\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1}) How to calculate [1/1] for f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-... ? Homework Equations...
10. ### Prove identity

It looks a bit for me like this theorem http://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html

12. ### Prove identity

Sorry. I made a mistake. You can see know what I meant. I edit my last message.
13. ### Prove identity

For example \frac{d}{d\alpha}f(\alpha^ax)=ax\alpha^{a-1}\frac{\partial f}{\partial \alpha} Right?
14. ### Prove identity

Homework Statement \alpha\frac{d}{d\alpha}[f(\alpha^ax,\alpha^by)]|_{\alpha=1}=ax\frac{\partial f}{\partial x}+by\frac{\partial f}{\partial y} Homework Equations The Attempt at a Solution Homework Statement I'm confused. I don't know what to do here. How to differentiate left...
15. ### Find derivative

Homework Statement \frac{d^{2n}}{dx^{2n}}\cos x n \in N Homework Equations \cos x=\sum^{\infty}_{k=0}(-1)^k\frac{x^{2k}}{(2k)!} The Attempt at a Solution \frac{d^{2n}}{dx^{2n}}x^{2n}=(2n)! But k is different that n. I don't have a clue how to solve that.
16. ### Derivative homework help

Homework Statement Calculate \lim_{h\to 0}\frac{F[p(x)+hp'(x)]-F[p(x)]}{h} where F'=f Homework Equations \lim_{h\to 0}\frac{F(x+h)-F(x)}{h}=F'(x) The Attempt at a Solution I think that solution is p'(x)f[p(x)] but I have a trouble to get the result.
17. ### Prove identity

Homework Statement \frac{\partial \vec{r}}{\partial q_i}gradf(r)=\frac{\partial f(r)}{\partial q_i} Homework Equations gradf(r)=\frac{df}{dr}gradr=\frac{df}{dr}\frac{ \vec{r}}{r}=\frac{df}{dr}\vec{r}_0 The Attempt at a Solution \frac{\partial \vec{r}}{\partial...
18. ### Solve ODE y''-y=e^{-t}

How you choose form of particular solution?
19. ### Solve ODE y''-y=e^{-t}

How do you know how to look for the function?
20. ### Solve ODE y''-y=e^{-t}

How to find particular integral?
21. ### Solve ODE y''-y=e^{-t}

Homework Statement Solve ODE y''-y=e^{-t} y(0)=1, y'(0)=0 Homework Equations The Attempt at a Solution Homogenuous solution t^2-1=0 y=C_1e^t+C_2e^{-t} From y(0)=1, y'(0)=0 y=\frac{1}{2}e^t+\frac{1}{2}e^{-t} How from that get complete solution?
22. ### One hard integral

And how you get \int^{\infty}_0 \frac{\sin x}{\sqrt{x}}e^{-a x}dx=\sqrt{\frac{\pi}{2(a^2+1)(a+\sqrt{a^2+1})}}
23. ### One hard integral

Is there some other way?
24. ### One hard integral

Ok. But I would like to see some trick how to calculate this integral in the desert island :)
25. ### One hard integral

But t goes into \infty so I write in correct way last step.
26. ### One hard integral

Mistake I will get 2\int^{\infty}_0\sin v^2dv
27. ### One hard integral

Homework Statement Solve \int^{\infty}_0\frac{\sin x}{\sqrt{x}} Homework Equations The Attempt at a Solution \lim_{t\to \infty}\int^{t}_0\frac{\sin x}{\sqrt{x}} \sqrt{x}=v so \lim_{t\to \infty}\int^{t}_0\frac{\sin v^2}{v} Integration by parts maybe? What is idea?
28. ### F. transform problem

Homework Statement Find Fourier transform of function f(x)=\frac{1}{x^2+a^2}, a>0 Homework Equations \mathcal{F}[\frac{1}{x^2+a^2}]=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ikx}dx}{x^2+a^2} The Attempt at a Solution Two different case k>0 and k<0...
29. ### Integration by parts

Homework Statement Solve integral \int^{1}_0(1-x)\frac{d}{dx}\frac{\sin Cx}{C}dx Homework Equations \int udv=uv-\int vdu The Attempt at a Solution u=1-x dv=\frac{d}{dx}\frac{\sin Cx}{C}dx What is v? How to integrate \frac{d}{dx}\frac{\sin Cx}{C}dx?
30. ### Improper integrals with i

I don't know that equivalence. How can I show that this is equivalent?