# complex analysis question

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• Mar 25th 2011, 04:05 PM
shaheen7
complex analysis question
one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.
• Mar 25th 2011, 04:58 PM
HallsofIvy
Quote:

Originally Posted by shaheen7
one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.

The question is either trivial or doesn't make any sense. (z+ 3)^2+ i is a polynomial and all polynomials are differentiable for all z. On the other hand, "holomorphic" means "analytic for all z" so it makes no sense to ask for points on which a function "is holomorphic".
• Mar 25th 2011, 05:56 PM
shaheen7
For the holomorphic part won't we need to use the Cauchy-Riemann equations? How would we go about finding du/dx, dv,dy and dv,dx, -dv,dy
• Mar 25th 2011, 10:27 PM
chisigma
Quote:

Originally Posted by shaheen7
one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.

For semplicity sake we set $\displaystyle s=z+3$ and neglect the constant term i so that the function becomes $\displaystyle h(s)=s^{2}$. Setting $\displaystyle s=\sigma + i\ \omega$ You find that is...

$\displaystyle h(s) = u(\sigma, \omega) + i\ v(\sigma, \omega) = \sigma^{2}- \omega^{2} + 2\ i\ \sigma\ \omega$

... and now it is easy to verify that the Cauchy Riemann equations are satisfied for any value of s...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$