Taylor Series: Can't quite work it out

[wizard147]

Hi Guys,

Looking at some notes i have on conformal mapping and I have the following

where z is complex and z* denotes its conjugate, R is a real number


z* = -iR + R^2/(z-iR)

and my lecturer says that using the taylor series we get,

z* = -iR + iR(1+ z/iR + ...)

I've been trying for ages but I can't get this, I'm probably doing something stupid.

Anybody point me in the right direction? I'm getting confused with all these expansions!

I'm doing the following:

z* = -iR + R^2/z(1-iR/z)

and using the formula (1-x)^-1 = 1 + x^2 + x^3 (reference to wiki)

but it's not quite working! Hope you can help

C

 

[tiny-tim]

hi wizard147!

(try using the X² button just above the Reply box )

z* = -iR + R²/(z-iR)

and my lecturer says that using the taylor series we get,

z* = -iR + iR(1+ z/iR + ...)

you just have to fiddle around with it a little …

z* = -iR + R²/(z-iR)

= -iR + R/(z/R - i)

= -iR + iR/(z/iR - 1) …​

hmm, there's a sign wrong somewhere

 

[wizard147]

Hi Tim,

Thanks, I think when you factored out your i, and then multiplied top and bottom by i you would get

-iR + iR/(1-z/iR)

I could be wrong though! lol

 

[tiny-tim]

yup! …

i think i got confused about whether the last "i" was on the top or the bottom of iR/z/iR ! ​

thanks!