Chem Differential Eq HW Solutions Fall 2011 111

Chem Differential Eq HW Solutions Fall 2011 111 - | f ( x-t...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 7.2 The Fourier Transform 111 (c) With the help of Theorem 3 f * g = F - 1 ± - i w 2 e - 3 w 2 4 ² = F - 1 ± i 1 2 4 6 d dw e - 3 w 2 4 ² = 1 2 2 3 F - 1 ± i d dw e - 3 w 2 4 ² = 1 2 2 3 x F - 1 ± e - 3 w 2 4 ² = 2 3 3 xe - 1 3 x 2 . In computing F - 1 ± e - 3 w 2 4 ² , use Exercise 10(a) and (5) to obtain F - 1 ± e - aw 2 ² = 1 2 a e - ( - x ) 2 4 a = 1 2 a e - x 2 4 a . 57. Recall that f is integrable means that ³ -∞ | f ( x ) | dx < . If f and g are integrable, then f * g ( x )= 1 2 π ³ -∞ f ( x - t ) g ( t ) dt. So, using properties of the integral: ³ -∞ | f * g ( x ) | dx = ³ -∞ ´ ´ ´ ´ 1 2 π ³ -∞ f ( x - t ) g ( t ) dt ´ ´ ´ ´ dx 1 2 π ³ -∞ ³ -∞
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: | f ( x-t ) g ( t ) | dxdt (Interchange order of integration. 1 2 - = - | f ( x ) | dx - | f ( x-t ) | dx | g ( t ) | dt (Change variables in the inner integral X = x-t. ) 1 2 - | f ( x ) | dx - | g ( t ) | dt &lt; ; thus f * g is integrable....
View Full Document

Ask a homework question - tutors are online