Unformatted text preview: and the theorem in section 21. 8. Determine where f ( z ) = z 2 + 2 z + ¯ z 2 is diﬀerentiable and evaluate f ( z ). 9. Is the function f ( z ) = | z | 2 diﬀerentiable at 0? Is f analytic at 0? Show your work and explain. 10. Show that u ( x,y ) = xe x cos x-ye x sin y is harmonic and ﬁnd a harmonic conjugate of u . 11. Suppose f ( z ) = u ( x,y )+ iv ( x,y ) is analytic in a domain D . Find a harmonic conjugate w for v in D so that g ( z ) = v ( x,y ) + iw ( x,y ) is analytic in D . 12. Determine where f ( z ) = zImz is diﬀerentiable and evaluate f ( z ). Do this two ways: (a) Using the deﬁnition of the derivative; (b) Using the Cauchy-Riemann Equations. 13. Describe the behavior of e z = e x e iy as (a) x → -∞ ; (b) y → ∞ . 14. Evaluate log(-1-i ) and Log (-1-i )....
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- Spring '08
- Derivative, Holomorphic function, Harmonic conjugate