Mathematical Physics Vol 1

6.3 Examples

329

Problem 226 Expand the function f ( x )= x sin x into a Fourier series in the interval ( − π , π ) .

Result

∞ ∑ n = 2

( − 1 ) n + 1 n 2 − 1

1 2

f ( x )= 1 −

cos x + 2

cos nx .

Problem 227 Expand the function f ( x )= x cos x into a Fourier series in the interval − π 2 , π 2 .

Result

∞ ∑ n = 1

( − 1 ) n + 1 n ( 4 n 2 − 1 ) 2

16 π

f ( x )=

sin2 nx .

Problem 228 Expand the function f ( x )= x ( π − x ) into a Fourier series in the interval [ 0 , π ) , f ( x )= f ( x + π ) .

Result

∞ ∑ n = 1

π 2

cos2 nx n 2

f ( x )=

6 −

.

Problem 229 Expand the function f ( x )=

π − x 2

into a Fourier series in the interval ( 0 , 2 π ) .

Result

∞ ∑ n = 1

sin nx n

f ( x )=

.