Does anyone know the sum of Fourier series sum_(m=0)^infty (cos(2m+1)x)/(2m_1) ?

Nicholas Hunter

Nicholas Hunter

Answered question

2022-11-06

Does anyone know the sum of Fourier series
m = 0 cos ( 2 m + 1 ) x 2 m + 1 ?

Answer & Explanation

sellk9o

sellk9o

Beginner2022-11-07Added 11 answers

Rewrite the series as
[ n = 0 ( e i x ) 2 n + 1 2 n + 1 ] = [ arctanh ( e i x ) ] = 1 2 [ log ( 1 + e i x 1 e i x ) ]
With some manipulation, noting that log z = log | z | + i 2 π arg z , we find that
n = 0 cos ( 2 n + 1 ) x 2 n + 1 = 1 2 log | cot x 2 |

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