- #1

fishturtle1

- 394

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## Homework Statement

Suppose z = x + iy. Where are the following functions diﬀerentiable? Where are they holomorphic? Which are entire?

the function is f(z) = e

^{-x}e

^{-iy}

## Homework Equations

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

## The Attempt at a Solution

f(z) = e

^{-x}e

^{-iy}

I convert it to polar form:

f(z) = cos(xy) + isin(xy)

then i set u as the real part of the equation, and v as the complex part of the equation.

u = cos(xy)

v = isin(xy)

∂u/∂x = -ysin(xy)

∂v/∂y= ixcos(xy)

then ∂u/∂x =/= ∂v/∂y so the function is not holomorphic.

I think i did this wrong because I've done this for the majority of the problem set, and every function is coming up as not holomorphic. Is my work correct?

**EDIT:**I did it again and here is my new work, my other work was wrong because i multiplied exponents wrong

f(z) = e

^{-x}e

^{-iy}

= (cos(-x) + isin(-x)) * (cos(-y) + isin(-y))

= cos(-x)cos(-y) + icos(-y)sin(-x) + (-1)sin(-x)(sin-y) + isin(-x)cos(-y)

so Re(f(z)) = cos(-x)cos(-y) - sin(-x)sin(-y) = u

Im(f(z)) = cos(-y)sin(-x) + cos(-x)sin(-y) = v

then u

_{x}= -(-sin(-x)cos(-y)) + cos(-x)sin(-y)

u

_{x}= sin(-x)cos(-y) + cos(-x)sin(-y)

and v

_{y}= - (-sin(-y)sin(-x)) + cos(-x)cos(-y)

v

_{y}= sin(-y)sin(-x) - cos(-x)cos(-y)

so u

_{x}=/= v

_{y}

so f(z) is not holomorphic, and is not differentiable at all points( but may be differentiable at some points ), and is not entire because it is not differentiable at all points.

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