The Fourier series of a real periodic function has only :
P : cosine terms if it is even
Q : sine terms if it is odd
R : cosine terms if it is odd
S : sine terms if it is even
Which of the above statements are correct ?
MH SET - 2018, SET Exam - 2018,
Answer: B
The correct statements are P and Q.
An even function satisfies \(f(-t) = f(t)\). The sine terms are odd, so their coefficients \(b_{n}\) all vanish and the series contains only cosine terms (and possibly a constant). An odd function satisfies \(f(-t) = -f(t)\); now the cosine coefficients \(a_{n}\) and the constant vanish, and the series contains only sine terms.
Statements R and S reverse these rules: an odd function cannot be built from cosines, which are even, and an even function cannot be built from sines, which are odd. So any option including R or S is wrong.
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