Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

328

Problem 222 Expand the function f ( x )= sin ax into a Fourier series in the interval ( − π , π ) , ( a̸ = 0 , ± 1 , ± 2 ,... )

Result

∞ ∑ n = 1

( − 1 ) n + 1 n sin nx n 2 − a 2 .

2sin π a π

f ( x )=

Problem 223 Expand the function f ( x )= sh ax into a Fourier series in the interval ( − π , π ) .

Result

∞ ∑ n = 1

( − 1 ) n + 1 n sin nx n 2 + a 2 .

2sh π a π

f ( x )=

Problem 224 Expand the function f ( x )= ch ax into a Fourier series in the interval ( − π , π ) .

Result

π "

a cos nx a 2 + n 2 #

∞ ∑ n = 1

2sh a π

1 2 a

( − 1 ) n

f ( x )=

+

.

Problem 225 Expand the function f ( x )= e ax into a Fourier series in the interval ( − l , l ) .

Result

f ( x )= 2sh ( al ) · "

al cos nx − π n sin nx ( al ) 2 +( π n ) 2 # .

∞ ∑ n = 1

1 2 al

( − 1 ) n

+