MATH253 Chapter Notes - Chapter 1-24: Residue Theorem, Improper Integral, 4Dx

• School: University of Alberta
• Department: Mathematics
• Course: MATH253
• Professor: Souvik Goswami

Document Summary

Notes reproduced from james lewis"s book, with permission from the author. (i) improper integrals - cauchy principal values. Given a real function y = f (x), we interpret the improper integral (cid:90) (cid:90) r. We call this the cauchy principal value, and it is sometimes denoted by. For notational convenience, we will drop pv from our notation. Suppose y = f (x) is odd, i. e. f ( x) = f (x) [e. g. Suppose y = f (x) is odd, i. e. f ( x) = f (x) [e. g. sin x, x1, x3, x5, . Then f (x)dx = 0, hence f (x)dx = 0. (cid:90) r. Note that if e. g. f (x) = x, then(cid:90) (cid:90) 0 (cid:90) and so and. 0 (cid:90) xdx = , xdx = , (cid:90) 0 (cid:90) . 0 x2 + 1 (cid:90) dx = 1 is meaningless, even though the cauchy pv makes sense and is de ned!