Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
Hankel transform: K…p; x† ˆxJ n …px†; a ˆ 0; b ˆ1, where J n …px† is the
Bessel function of the ®rst kind of order n:
H‰ f …x†Š ˆ
Z 1
0
f …x†xJ n …x†dx ˆ F…p†:
Mellin transform: K…p; x† ˆx p1 ,anda ˆ 0; b ˆ1:
M‰ f …x†Š ˆ
Z 1
0
f …x†x p1 dx ˆ F…p†:
The Laplace transform has been the subject of this chapter, and the Fouier
transform was treated in Chapter 4. It is beyond the scope of this book to include
Hankel and Mellin transformations.
Problems
9.1 Show that:
(a) e t2 is not of exponential order as x !1.
(b) sin e t2 is of exponential order as x !1.
9.2 Show that:
a
(a) L‰sinh axŠ ˆ
p 2 a 2 ; p > 0:
(b) L‰3x 4 2x 3=2 ‡ 6Š ˆ72
p 5 3 p
2p ‡ 6 5=2 p .
(c) L‰sin x cos xŠ ˆ1=…p 2 ‡ 4†:
(d) If
f …x† ˆ x; 0 < x < 4 ;
5; x > 4
then
L‰ f …x† ˆ 1
p 2 ‡ e4p
p
e4p
p 2 :
9.4 Show that L‰U…x a†Š ˆ e ap =p; p > 0:
9.5 Find the Laplace transform of H…x†, where
H…x† ˆ x;
5;
0 < x < 4
:
x > 4
9.5 Let f …x† be the recti®ed sine wave of period P ˆ 2:
sin x; 0 < x <
f …x† ˆ
0; x < 2 :
Find the Laplace transform of f …x†.
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