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# 374bhw9 - HW#9 Phys374Spring 2007 Due before class Friday...

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HW#9 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Friday, April 13, 2007 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ [email protected] 1. Express the real and imaginary parts of the following functions in terms of x = Re ( z ) and y = Im ( z ): z 3 , e z , e iz , sin z , 1 / ( z 2 + 1). 2. Evaluate ( z 2 - z ) dz along the following two contours connecting 0 to 1 + i : (a) from 0 to 1 along the real axis and then 1 to 1 + i along the imaginary direction; (b) along the diagonal directly from 0 to 1 + i . (c) Verify that you obtain the same result either way, and explain how you could have known the two integrals would be the same without even evaluating them. 3. Find the residues of the following functions at the given values of z : (a) ( z + z 2 ) - 1 at 0 and at - 1. (b) ln(1 + 2 z ) /z 2 at 0 (c) [ z 3 ( z + 2) 2 ] - 1 at 0 and at - 2. (d) cos z/ (2 z - π ) 4 at π/ 2 (e) ( z 2 + 1) - 3 at ± i . Hint : See the supplement for methods of evaluating residues. 4. Consider the real integral -∞ dx ( x 2 + a 2 )( x 2 + b 2 ) where a and b are positive real numbers.

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374bhw9 - HW#9 Phys374Spring 2007 Due before class Friday...

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