Series converges to a function that satisfies the wave equation

[Fogs]

Problem: show that the series \sum(1/n^2)*sin(nx)*exp(-ny) converges to a continuous function u(x,y),

Then show that U satisfies Uxx + Uyy = 0

Attempt: By the M-test, I know it converges, but I have to find the function it converges to. I tried to simplify the sum by using an identity (euler's), but it didn't make sense because this is not complex. I know the solution should look something like: (1/2)*(f(x+cy)+f(x-cy)), but I don't know how to get there. please help! I've been trying to figure this out for hours!

 

[tiny-tim]

Welcome to PF!

Hi Fogs! Welcome to PF!

(try using the X² and X₂ icons just above the Reply box )

from https://www.physicsforums.com/library.php?do=view_item&itemid=162" you should get sinx = i(e^-ix - e^ix)/2

 

[Fogs]

Thanks!