How can e^(1/z) be written using the definition of e^z?

[filter54321]

How do you write e^(1/z) in the other form?

z = x+yi

So we should be able to right it using this definition of e^z, no?

e^z = e^x * [cos(y) + i * sin(y)]

I pushed some numbers around the page for a while but I can't get 1/(x+i*y) to split into anything nice. Is there a trick?

 

[filter54321]

...actually, I think I was confusing myself. The x,y in the definition don't have to be matching up with the x,y from the complex number z. The x,y in the definition correspond to the real and imaginary parts of an arbitrary complex z.Would you plug this into the definition?

real:
x/(x^2+y^2)

imaginary:
-y/(x^2+y^2)

 

[Liquidxlax]

...actually, I think I was confusing myself. The x,y in the definition don't have to be matching up with the x,y from the complex number z. The x,y in the definition correspond to the real and imaginary parts of an arbitrary complex z.


Would you plug this into the definition?

real:
x/(x^2+y^2)

imaginary:
-y/(x^2+y^2)

simple answer no.

z=x+iy

[Re]z = x [Im]z = y

or z= cosq +isinq => e^iq

if you look in your textbook you should have all the definitions necessary

 

[gomunkul51]

e^(1/z)
z = x + iy = e^itheta = cos(theta) + isin(theta)

e^(1/z) = e^(1/e^itheta) = e^(e^i(-theta)) = e^(cos(-theta) + isin(-theta)) = ...

can you go from here?