Page 373 - Schaum's Outline of Theory and Problems of Advanced Calculus

364 FOURIER INTEGRALS [CHAP. 14

In the generalization of Fourier coefficients to Fourier integrals, a 0 may be neglected, since whenever
ð
1
f ðxÞ dx exists,
1
ð L
1

f ðxÞ dx ! 0 as

ja 0 j¼ L !1

L L
EQUIVALENT FORMS OF FOURIER’S INTEGRAL THEOREM
Fourier’s integral theorem can also be written in the forms
1 ð 1 ð 1
f ðuÞ cos ðx uÞ du d
f ðxÞ¼ ¼0 u¼ 1 ð3Þ
1 ð 1 i x ð 1 i u
e d f ðuÞ e du
f ðxÞ¼ ð4Þ
1
2 1
1 ð 1 ð 1
f ðuÞ e i ðu xÞ du d
¼
1
2 1
where it is understood that if f ðxÞ is not continuous at x the left side must be replaced by
f ðx þ 0Þþ f ðx 0Þ
.
2
These results can be simplified somewhat if f ðxÞ is either an odd or an even function, and we have
2 ð 1 ð 1
cos xd f ðuÞ cos udu if f ðxÞ is even
f ðxÞ¼ ð5Þ
0 0
2 ð 1 ð 1
sin xd f ðuÞ sin udu if f ðxÞ is odd
f ðxÞ¼ ð6Þ
0 0
An entity of importance in evaluating integrals and solving differential and integral equations is
introduced in the next paragraph. It is abstracted from the Fourier integral form of a function, as can
be observed by putting (4)in the form
1 ð 1 i x 1 ð 1 i u
e e f ðuÞ du d
f ðxÞ¼ p ffiffiffiffiffiffi p ffiffiffiffiffiffi
2 1
2 1
and observing the parenthetic expression.

FOURIER TRANSFORMS
From (4)it follows that
1 ð 1 i u
f ðuÞ e du
Fð Þ¼ p ffiffiffiffiffiffi ð7Þ
2 1
1 ð 1 i x
then f ðxÞ¼ p ffiffiffiffiffiffi Fð Þ e d (8)
2 1
The function Fð Þ is called the Fourier transform of f ðxÞ and is sometimes written Fð Þ¼ ff f ðxÞg.
1
The function f ðxÞ is the inverse Fourier transform of Fð Þ and is written f ðxÞ¼ f fFð Þg.
p ffiffiffiffiffiffi
Note: The constants preceding the integral signs in (7) and (8) were here taken as equal to 1= 2 .
However, they can be any constants different from zero so long as their product is 1=2 . The above is
called the symmetric form. The literature is not uniform as to whether the negative exponent appears in
(7)or in(8).