Complex Logarithm Homework: Find -Ln(1-e(iθ))

[soothsayer]

Homework Statement

find -Ln(1-e^{(i[tex]\theta[/tex]}) (in terms of theta)

(this is me just skipping the part of the problem I know and going straight to what I can't figure out)

Homework Equations

ln(z) = ln(re^{i[tex]\theta[/tex]})=Ln(r) + ln(e^{i[tex]\theta[/tex]}) = Ln(r) + i[tex]\theta[/tex]

The Attempt at a Solution

I don't really know how to break this logarithm up into real and complex parts, the two ways I considered were

= -(Ln(1-1) + i[tex]\theta[/tex])
but that ends up with Ln(0) which blows up to infinity and doesn't make sense in this problem.

= -(Ln(1) + i[tex]\theta[/tex])
but this is just -i[tex]\theta[/tex] which leaves only an imaginary part, and the problem implies there is a real logarithmic solution as well.

 

[tiny-tim]

hi soothsayer!

(have a theta: θ )

you need to express 1 - e^iθ in Cartesian form, a + ib