Unformatted text preview: Suggestion: Consider f 2 .) 7. Let f = u + iv be analytic (in some suitable region). In each of the following, ﬁnd v given u . (i) u = 2 x 2 + 2 x + 1-2 y 2 (ii) u = y x 2 + y 2 8. Suppose that D is a region. Show that there are no analytic functions f = u + iv on D with u ( x,y ) = x 2 + y 2 . 9. Suppose f is an entire function satisfying the diﬀerential equation f ( z ) = αf ( z ) for every complex number z (where α is a ﬁxed nonzero complex constant). Prove that f ( z ) = C exp( αz ), for some complex constant C . ( Suggestion: Consider the function g ( z ) := exp(-αz ) f ( z ) and its derivative.) 10. Let f ( z ) = exp( z ), z ∈ C . Give a direct proof of the fact that f ( z ) = exp( z ). 11. Verify that the functions sin z and cos z are entire. Show that the derivative of sin z is cos z , and that the derivative of cos z is-sin z . 1...
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- Fall '08
- Holomorphic function, Principal value, Sivakumar Example Sheet, nonconstant analytic function