Chem Differential Eq HW Solutions Fall 2011 124

Chem Differential Eq HW Solutions Fall 2011 124 - = 1 2 ²...

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

View Full Document Right Arrow Icon
124 Chapter 7 The Fourier Transform and its Applications Solutions to Exercises 7.6 1. The even extension of f ( x )is f e ( x )= ± 1i f - 1 <x< 1, 0 otherwise. The Fourier transform of f e ( x ) is computed in Example 1, Sec. 7.2 (with a = 1). We have, for w 0, F c ( f )( w )= F ( f e )( w )= ² 2 π sin w w . To write f as an inverse Fourier cosine transform, we appeal to (6). We have, for x> 0, f ( x )= ² 2 π ³ 0 F c ( f )( w ) cos wxdw, or 2 π ³ 0 sin w w cos wxdw = 1i f 0 <x< 1, 0i f x> 1 , 1 2 if x =1 . Note that at the point x = 1, a point of discontinuity of f , the inverse Fourier transform is equal to ( f ( x +) + f ( x - )) / 2. 5. The even extension of f ( x )is f e ( x )= ± cos x if - 2 π<x< 2 π , 0 otherwise. Let’s compute the Fourier cosine transform using de±nition (5), Sec. 7.6: F c ( f )( w )== ² 2 π ³ 2 π 0 cos x cos wxdx = ² 2 π ³ 2 π 0 1 2 [cos( w +1) x + cos( w - 1) x ] dx = 1 2 ² 2 π ´ sin( w +1) x w +1 + sin( w - 1) x w - 1 µ¶ 2 π 0 ( w ± =1)
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = 1 2 ² 2 π ´ sin 2( w + 1) π w + 1 + sin 2( w-1) π w-1 µ ( w ± = 1) = 1 2 ² 2 π ´ sin 2 πw w + 1 + sin 2 πw w-1 µ ( w ± = 1) = ² 2 π sin2 πw w w 2-1 ( w ± = 1) . Also, by l’Hospital’s rule, we have lim w → ² 2 π sin 2 πw w w 2-1 = √ 2 π, which is the value of the cosine transform at w = 1. To write f as an inverse Fourier cosine transform, we appeal to (6). We have, for x > 0, 2 π ³ ∞ w w 2-1 sin 2 πw cos wxdw = ± cos x if 0 < x < 2 π , i f x > 2 π. For x = 2 π , the integral converges to 1 / 2. So 2 π ³ ∞ w w 2-1 sin2 πw cos 2 πwdw = 1 2 ....
View Full Document

This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

Ask a homework question - tutors are online