How would you integrate this function?

[Vitani11]

Homework Statement

I need to find the coefficients for a Fourier expansion. Here is the integral I need to solve: (1/π) ∫sin(x/2)sin(nx)dx where the limits are from -π to π. The original function for the expansion is f(x) = sin(x/2)

Homework Equations

None

The Attempt at a Solution

Should I do this in terms of complex numbers? I think that should be my approach but I am not sure. If this is the right approach can you help me get started with setting up the integral?

 

[Mark44]

Homework Statement

I need to find the coefficients for a Fourier expansion. Here is the integral I need to solve: (1/π) ∫sin(x/2)sin(nx)dx where the limits are from -π to π. The original function for the expansion is f(x) = sin(x/2)

Homework Equations

None

The Attempt at a Solution

Should I do this in terms of complex numbers?

I don't think so. The usual approach is integration by parts, twice. Here's a link to a similar example, https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintsoldirectory/TrigIntSol3.html#SOLUTION 24, problem #24

I think that should be my approach but I am not sure. If this is the right approach can you help me get started with setting up the integral?

 

[Dr Transport]

use the identity

$$\frac{1}{2\pi} \int_{-\pi} ^{\pi} \sin(nx)\sin(mx) dx = \delta_{nm}$$

and the integral pops right out for you . (i might have the normalization wrong... off the top of my head...)

 

[Vitani11]

Woah, so you're saying that using the kronecker delta I can just say it is δ_{n (1/2)}? Can you walk me through those steps? Does it have something to do with the dirac delta function/identities? If the kronecker truly makes it simpler then I'd rather use that lol.

 

[Dr Transport]

it is a definition...if you write the sine functions in terms of complex exponential's and do the integrals, it comes out very quickly...

 

[Dr Transport]

$$\frac{1}{\pi}\int_{-\pi}^{\pi}\sin(nx)\sin(mx)dx = \delta_{nm}$$

 

[Vitani11]

Yes I've solved this now- thank you