Solution to Fourier Transform of f(θ) = |sin(θ)|

[Stu165]

I need to show the solution to the Fourier transfor of f(theta) = |sin(theta)|.

However i think that solving this needs to be done by complex anaylsis as integration by parts just keeps going on and on.
Does anyone know where to go with this?

 

[Hurkyl]

However i think that solving this needs to be done by complex anaylsis as integration by parts just keeps going on and on.

Does it provide an equation you can solve? Integrating \int e^x \sin x \, dx is a standard IBP problem -- I bet it's in your calc 2 text.

 

[Stu165]

I need to find

1/pi int sin(theta)cos(mk theta) d theta from 0 - pi

pi being the period

excuse the type I don't know how to do the equation thing.

I only find the cosine part of the transform cause it's and even function, therefore I don't integrate any sine components at the start. If that makes sense.

 

[Hurkyl]

Sorry, I'm used to the exponential version.

All of your problems are solved with a simple trigonometric identity -- you can change a product of sine and cosine into a sum of two sines. I don't remember the exact form, but you can derive it yourself. Hint: consider sin(A+B) + sin(A-B).

(But, as you guessed, you could also do this by replacing cos x with Re[e^(ix)])

 

[whozum]

sin(2x) = 2 sinx cosx but in this case sin and cosin have different parameters. I don't know how you'll go about that.

 

[Hurkyl]

Hint: consider sin(A+B) + sin(A-B)

 

[Stu165]

forgot to look at trig identities yet. I'll try that