Unformatted text preview: √ π/ 2. 2. (30 pts.) Evaluate the following integrals using contour integration: (a) (10 pts.) i ∞ dx (1 + x 2 ) 2 , (b) (20 pts.) i ∞ dx 1 + x 3 . [Hint for (b): use the boundary of a 120 ◦ circular sector of large radius as integration contour] 3. (30 pts.) Show that (a) (15 pts.) i ∞ (ln x ) 2 1 + x 2 dx = π 3 8 , (b) (15 pts.) i ∞ ln x 1 + x 2 dx = 0 . [Hint: Obtain both integrals simultaneously by integrating some branch of (log z ) 2 / (1 + z 2 ) around a large semicircle indented at z = 0.] 4. (20 pts.) Evaluate i ∞ x − α 1 + x 4 dx for α real. What are the restrictions on real α for this integral to exist? Due at 5pm on Wed. November 25....
View
Full Document
 Fall '09
 PROF
 Calculus, pts

Click to edit the document details