I What is the closed form expression for f(a,b,n)?

[MAGNIBORO]

hi, I do not know much about PDEs and programs like wolfram alpha and maple don't give me a solution.
it is possible to calculate the function through PDE?.
I would appreciate any help

$$\frac{\partial }{\partial a}f(a,b,n)+\frac{\partial }{\partial b}f(a,b,n)=-n f(a,b,n+1)$$

$$f(a,b,0)=\frac{\pi }{2} \: \; \, \,, \, \, \, f(a,b,1)=\frac{\pi }{2\sqrt{a b}}$$as we know $f(a,b,1)$ We can calculate $f(a,b,2)\,,\,f(a,b,3),...$

But we could calculate closed form expression for $f(a,b,n)$ ?

thanks

 

[MAGNIBORO]

i reduce the problem to solve
$$\frac{\partial }{\partial c} \, g_{n}(c) = -n \, g_{n+1}(c)$$
with
$$g_{1}(c)=\frac{\pi }{2\sqrt{c^{2}+c}}$$

 

[Strum]

Do you need the result for all c?

 

[MAGNIBORO]

Do you need the result for all c?

i found the solution, The function was the result of a definite integral.
thanks anyway

 

[Strum]

Well if you found the solution I would like to see it. Please post it :)

 

[MAGNIBORO]

the function is:
$$\int_{0}^{\frac{\pi }{2}}\left ( a \, cos^2(x) +b \, sin^2(x) \right )^{-n}$$

 

[Strum]

Ehh but this is not dependent on c as specified in post number 2.

 

[MAGNIBORO]

evaluate the function as $f(c,c+1,n)$ and check