Finding the Fourier series of a function.

[bubokribuck]

Homework Statement

f(x)=
-cos(x) when -π<x<0
cos(x) when 0<x<π

Decide if f is an even, odd function or either.
Find the Fourier series of f.

Homework Equations

odd function: f(x)=f(-x)
even function: -f(x)=f(-x) or f(x)=-f(-x)

The Attempt at a Solution

substitute -x into either cos(x) or -cos(x) => -cos(x)=-cos(-x) and cos(x)=cos(-x),
therefore, f is an even function.

However, I'm stuck when it comes to finding the Fourier series.

I know how to solve a₀, where I just need to find the integration of -cos(x)dx and cos(x)dx. To find a_n and b_n, I need to find the integration of [-cos(x)cos(nx)dx], [cos(x)cos(nx)dx], [-cos(x)sin(nx)] and [cos(x)sin(nx)dx]. I tried to solve them using integration by parts, but it turned out to be infinitely expanding, so I guess integration by parts won't work. Is there any other way to integrate the above four functions?

 

[Ray Vickson]

Homework Statement

f(x)=
-cos(x) when -π<x<0
cos(x) when 0<x<π

Decide if f is an even, odd function or either.
Find the Fourier series of f.

Homework Equations

odd function: f(x)=f(-x)
even function: -f(x)=f(-x) or f(x)=-f(-x)

The Attempt at a Solution

substitute -x into either cos(x) or -cos(x) => -cos(x)=-cos(-x) and cos(x)=cos(-x),
therefore, f is an even function.

However, I'm stuck when it comes to finding the Fourier series.

I know how to solve a₀, where I just need to find the integration of -cos(x)dx and cos(x)dx. To find a_n and b_n, I need to find the integration of [-cos(x)cos(nx)dx], [cos(x)cos(nx)dx], [-cos(x)sin(nx)] and [cos(x)sin(nx)dx]. I tried to solve them using integration by parts, but it turned out to be infinitely expanding, so I guess integration by parts won't work. Is there any other way to integrate the above four functions?

Your definitions of "even" and "odd" are the exact opposite of everybody else's in the world.

RGV

 

[LCKurtz]

I need to find the integration of [-cos(x)cos(nx)dx], [cos(x)cos(nx)dx], [-cos(x)sin(nx)] and [cos(x)sin(nx)dx]. I tried to solve them using integration by parts, but it turned out to be infinitely expanding, so I guess integration by parts won't work. Is there any other way to integrate the above four functions?

You need the product formulas:

http://www.sosmath.com/trig/prodform/prodform.html

 

[bubokribuck]

Your definitions of "even" and "odd" are the exact opposite of everybody else's in the world.

RGV

I just typed it wrong but they wouldn't let me edit it. :(

 

[bubokribuck]

You need the product formulas:

http://www.sosmath.com/trig/prodform/prodform.html

Thanks :)