Engineering Board Exam Reviewer · Fourier Series · easy
Which Fourier coefficients vanish for the odd function f(x)=x on (-π,π)?
- All bn
- Only a0
- All an, including a0
- Only the even-numbered bn
Answer
C. All an, including a0
Solution
For an odd function the cosine integrals -ππf(x) nx,dx vanish, so every an (including a0) is zero. Only the sine terms bn remain.
More Fourier Series questions
- The Fourier series of f(x)=x2 on (-π,π) is π2/3+4 n=1∞(-1)n/n2 nx. Evaluate it at x=π to find n=1∞ 1n2.
- The half-range sine series of f(x)=x on (0,2) is bn π x/2. Find b2.
- The square wave f(x)=-1 on (-π,0) and f(x)=1 on (0,π) has a jump at x=0. To what value does its Fourier series converge at x=0?
- Find a1 in the Fourier series of f(x)=|x| on (-π,π).
- Find the constant term a0/2 of the Fourier series of f(x)=x2 on (-π,π).
- The square wave f(x)=-1 on (-π,0) and f(x)=1 on (0,π) has Fourier coefficients bn. Find b3.
- Find b3 in the Fourier series of f(x)=x on (-π,π).