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I'm not too sure how to show this. Perhaps if I show d(e

^{z})/dz = e^{z}then does this conclude that e^{z}is holomorphic on all of C?
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- Thread starter Firepanda
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- #2

Dick

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I was told not to use limits here so I didn't want to use the base definition of holomorphic-ness.

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Because in my notes he mentioned the power series

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Dick

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I was told not to use limits here so I didn't want to use the base definition of holomorphic-ness.

Sure, use Cauchy-Riemann. That's easy enough. Then you'd want to show e^(-z) is also holomorphic. Do you know the sum of holomorphic functions is also holomorphic? If not then you could just directly show cosh(z) is holomorphic using CR.

- #6

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i tried to use C-R, but I'm unable to split it up into

U(x,y) or V(x,y)

using z=x+iy into cosh(z)

U(x,y) or V(x,y)

using z=x+iy into cosh(z)

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Dick

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i tried to use C-R, but I'm unable to split it up into

U(x,y) or V(x,y)

using z=x+iy into cosh(z)

How about e^z? Can you split that up?

- #8

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How about e^z. Can you split that up?

e

= e

=e

The C-R satisfy this and so it is holomorphic on all of C? so this would immediatly imply cosh(z) is entire?

And I would check if it were true for e

- #9

Dick

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Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.e^{x}e^{iy}

= e^{x}(cosy + i.siny)

=e^{x}cosy + i.e^{x}siny

The C-R satisfy this and so it is holomorphic on all of C? so this would immediatly imply cosh(z) is entire?

And I would check if it were true for e^{-z}as well.

- #10

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Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.

ah ye that would make more sense if i did it directly, thanks!

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