For f(x)=3sin(2x)+5cos x on [−π,π], find b₂ in the stated real Fourier convention.
Answer and explanation
3
Orthogonality removes the cosine term and all other sine modes.
Subjects
Fourier analysis
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Sample exercises
For the constant function f(x)=4, find a₀ in the convention f=a₀/2+... .
Answer and explanation
8
The constant term a₀/2 must equal 4.
At a jump from left limit 2 to right limit 6, what is the usual Fourier-series limit under piecewise-smooth hypotheses?
- 6
- 8
- 2
- 4
Answer and explanation
4
The limit is the midpoint of the one-sided limits.
What is the integral of sin(2x)sin(3x) from −π to π?
Answer and explanation
0
Distinct integer sine modes are orthogonal.
Explain why a finite Fourier sum cannot equal a discontinuous function pointwise everywhere.
Answer and explanation
Every finite sum of sines and cosines is continuous. A discontinuous target cannot be identical to a continuous function at every point.
Credit continuity of finite sums and the contradiction.
For a real odd function on [−π,π], which coefficients vanish?
- Every coefficient necessarily
- Only b₁
- a₀ and all cosine coefficients
- All sine coefficients
Answer and explanation
a₀ and all cosine coefficients
Odd integrands have zero symmetric integrals for the constant and cosine products.
Study notes
Fourier coefficients and convergence
On [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Advanced universityFourier coefficients and convergence · revision notes
On [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Advanced universitySine coefficient from orthogonality
This uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
Advanced universitySine coefficient from orthogonality · formula reference
This uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
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