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