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[http://www.math.purdue.edu/~bell/MA425/hwk2.txt HWK 2 problems] | [http://www.math.purdue.edu/~bell/MA425/hwk2.txt HWK 2 problems] | ||
| − | Here's a hint for II.3.1 (ii): | + | Here's a hint for II.3.1 (ii) --[[User:Bell|Bell]]: |
<math>\frac{f(z)g(z)-f(z_0)g(z_0)}{z-z_0}=\frac{f(z)g(z)-f(z)g(z_0)+f(z)g(z_0)-f(z_0)g(z_0)}{z-z_0}=</math> | <math>\frac{f(z)g(z)-f(z_0)g(z_0)}{z-z_0}=\frac{f(z)g(z)-f(z)g(z_0)+f(z)g(z_0)-f(z_0)g(z_0)}{z-z_0}=</math> | ||
<math>f(z)\frac{g(z)-g(z_0)}{z-z_0}+g(z_0)\frac{f(z)-f(z_0)}{z-z_0}.</math> | <math>f(z)\frac{g(z)-g(z_0)}{z-z_0}+g(z_0)\frac{f(z)-f(z_0)}{z-z_0}.</math> | ||
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| + | Here is a hint for II.8.1 (c) --[[User:Bell|Bell]]: | ||
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| + | If the modulus of <math>f=u+iv</math> is constant, then | ||
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| + | <math>u^2+v^2=c.</math> | ||
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| + | Take the partial derivative of this equation with respect to x to get one equation. Take it with respect to y to get another. Use the Cauchy-Riemann equations to conclude that the gradients of both u and v must be identically zero. | ||
Revision as of 10:44, 10 September 2009
Homework 2
Here's a hint for II.3.1 (ii) --Bell:
$ \frac{f(z)g(z)-f(z_0)g(z_0)}{z-z_0}=\frac{f(z)g(z)-f(z)g(z_0)+f(z)g(z_0)-f(z_0)g(z_0)}{z-z_0}= $
$ f(z)\frac{g(z)-g(z_0)}{z-z_0}+g(z_0)\frac{f(z)-f(z_0)}{z-z_0}. $
Here is a hint for II.8.1 (c) --Bell:
If the modulus of $ f=u+iv $ is constant, then
$ u^2+v^2=c. $
Take the partial derivative of this equation with respect to x to get one equation. Take it with respect to y to get another. Use the Cauchy-Riemann equations to conclude that the gradients of both u and v must be identically zero.
