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FOURIER INTEGRALS
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14.18. If f x
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1 0@x<1 nd the (a) Fourier sine transform, (b) Fourier cosine transform of f x . In 0 xA1 each case obtain the graph of f x and its transform. r  r  2 1 cos 2 sin ; b : Ans: a  
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14.19. (a) Find the Fourier sine transform of e x , x A 0 1 x sin mx  b Show that dx e m ; m > 0 by using the result in a : 2 x2 1 0 Ans. (c) Explain from the viewpoint of Fourier s integral theorem why the result in (b) does not hold for m 0. p (a) 2= = 1 2 8 <1 Y x sin xt dx 2 : 0
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14.20. Solve for Y x the integral equation 1
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0@t<1 1@t<2 tA2
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and verify the solution by direction substitution. Ans. Y x 2 2 cos x 4 cos 2x =x PARSEVAL S IDENTITY 14.21. Evaluate 1 dx x2 dx ; b by use of Parseval s identity. 2 2 2 2 0 x 1 0 x 1 [Hint: Use the Fourier sine and cosine transforms of e x , x > 0.] Ans: a =4; b =4 1 (a) (a)  1 1 cos x 2  dx ; x 2 0 1 b
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14.22. Use Problem 14.18 to show that 1 14.23. Show that
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sin4 x  dx . 2 x2
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x cos x sin x 2  dx . 15 x6
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MISCELLANEOUS PROBLEMS 14.24. (a) Solve @U @2 U 2 2 , U 0; t 0; U x; 0 e x ; x > 0; U x; t is bounded where x > 0; t > 0. @t @x (b) Give a physical interpretation. U x; t 2  1
Ans:
e 2 t sin x d 2 1 & x 0@x@1 , U x; t is bounded where x > 0; t > 0. 0 x>1
14.25. Solve
@U @2 U 2 ; Ux 0; t 0; U x; 0 @t @x U x; t 2  1
Ans:
 sin  cos  1 2 t cos x d e  2
14.26. (a) Show that the solution to Problem 14.13 can be written p p 2 x=2 t v2 1 1 x =2 t v2 e dv p e dv U x; t p  0  1 x =2p t
FOURIER INTEGRALS
[CHAP. 14
@U @2 U 2 and the conditions of Problem 14.13. @t @x & 1 jxj < 1 14.27. Verify the convolution theorem for the functions f x g x . 0 jxj > 1 (b) Prove directly that the function in (a) satis es 14.28. Establish equation (4), Page 364, from equation (3), Page 364. 14.29. Prove the result (12), Page 365. 1 1 f u ei u du and Hint: If F p 2 1 F G ! Now make the transformation u v x: 1 F G p  De ne 1 f g p  then 1 F G p  1 f u g x u du
1 G p 2 1 1
1 g v ei v dv,
then
1 2
ei u v f u g v du dv
1 1
1 1 ei x f u g x u du dx
1 1
f g is a function of x
1 1 ei x f g dx
1 1
Thus, F G is the Fourier transform of the convolution f g and conversely as indicated in (13) f g is the Fourier transform of F G . 14.30. If F and G are the Fourier transforms of f x and g x respectively, prove (by repeating the pattern of Problem 14.29) that 1 1 F G d f x g x dx
1 1
where the bar signi es the complex conjugate.
Observe that if G is expressed as in Problem 14.29 then 1 1 " " G e i x f u g v dv  1
14.31. Show that the Fourier transform of g u x is ei x , i.e., 1 1 i u ei x G p e f u g u x du  1 Hint: See Problem 14.29. Let v u x.
14.32. Prove Riemann s theorem (see Problem 14.10).
Gamma and Beta Functions
THE GAMMA FUNCTION The gamma function may be regarded as a generalization of n! (n-factorial), where n is any positive integer to x!, where x is any real number. (With limited exceptions, the discussion that follows will be restricted to positive real numbers.) Such an extension does not seem reasonable, yet, in certain ways, the gamma function de ned by the improper integral 1 x tx 1 e t dt 1
meets the challenge. This integral has proved valuable in applications. However, because it cannot be represented through elementary functions, establishment of its properties take some e ort. Some of the important ones are outlined below. The gamma function is convergent for x > 0. (See Problem 12.18, 12.) The fundamental property x 1 x x may be obtained by employing the technique of integration by parts to (1). The process is carried out in Problem 15.1. From the form (2) the function x can be evaluated for all x > 0 when its values in the interval 1 % x < 2 are known. (Any other interval of unit length will su ce.) The table and graph in Fig. 15-1 illustrates this idea.
_5 _4
(n)
5 4 3 2 1 _3 _2 _1 _1 _2 _3 _4 _5 1 2 3 4 5
TABLES OF VALUES AND GRAPH OF THE GAMMA FUNCTION n 1.00 1.10 1.20 1.30 n 1.0000 0.9514 0.9182 0.8975 375
Fig. 15-1