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exsheet5

# exsheet5 - Suggestion Consider f 2 7 Let f = u iv be...

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Sivakumar M407 Example Sheet 5 In what follows we shall use the following notation: H := { z = x + iy C : x 0 , y R } 1. Suppose 0 < α < 2 is a fixed number, and let f ( z ) denote the principal branch of the (multi- valued) function z 7→ z α . Find (and sketch) the image of the set H \ { 0 } under f . 2. Show that the function w = g ( z ) = exp( z 2 ) maps the lines x = - y and x = y onto the circle | w | = 1. Show further that g maps each of the two pieces of the set { z = x + iy : x 2 > y 2 } onto the set { w : | w | > 1 } and each of the two pieces of the set { z = x + iy : x 2 < y 2 } onto the set { w : | w | < 1 } . 3. (i) Find all values of the number i i . What is its principal value? (ii) Find all values of the number ( i i ) i . What is its principal value? (iii) What is the connexion between ( i i ) i and i i 2 ? What, if any, is the moral of this story? 4. Suppose that f is analytic on a region D . Prove that f is constant on D if and only if | f | is constant on D . 5. Show that a nonconstant analytic function cannot map a region onto an arc of a circle. 6. Assume f is analytic in a region and that at every point in the region, either f = 0 or f 0 = 0. Show that f is a constant. (
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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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