HW4 - MATH 185 SOLUTIONS OF HW IV(1(24 7(a f(z is real-valued in D = f(z = f(z By the main result in Ex 3 Sec 24 f is constant in D since f and f = f

HW4 - MATH 185 SOLUTIONS OF HW IV(1(24 7(a f(z is...

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MATH 185 SOLUTIONS OF HW IV (1) (24. 7) (a) f ( z ) is real-valued in D . = f ( z ) = f ( z ). By the main result in Ex 3, Sec 24, f is constant in D since f and f = f are both analytic in D . (b) If c = 0 then | f ( z ) | = 0, i.e. f ( z ) 0 in D . Hence f is constant. Otherwise f ( z ) = c 2 /f ( z ) is well-defined and analytic. By the main result in Ex 3, Sec 24, f is constant in D (2) (33.13) By using the formula (11),(12) on page 102, we can easily show that sin(¯ z ) = sin z and cos(¯ z ) = cos z . If sin(¯ z ) is analytic at some point p then there is an open neighborhood D of p such that sin(¯ z ) is analytic on D , i.e. sin z is analytic on D . By the main result in Ex 3, Sec 24, sin z is constant on D . But sin z is not constant in any open region, so this is contradiction, i.e. sin(¯ z ) is not analytic at any point. Similarly, cos(¯ z ) is not analytic. (3) (30.5) (a) i 1
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  • Fall '07
  • Lim
  • Math, Open set, main result

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