- #1

jaus tail

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## Homework Statement

A function is given as W = ln (Z)

Z = x + i y

W = u + i v

i is sqrt of (-1)

The u = constant lines get mapped in z plane as ?

## Homework Equations

Z = x + i y = [ sqrt {x

^{2}+ y

^{2}} ] e

^{i (theta)}

where theta is tan

^{-1}(y/x)

## The Attempt at a Solution

Function w is

w = ln (z)

= ln( sqrt {[x

^{2}+ y

^{2}] } multiply with e

^{i (theta)}

Now ln (a b) = ln (a) + ln (b)

so w = ln (sqrt [x

^{2}+ y

^{2}]) + i(theta) ln (e)

The first part is real, while second part is imaginary.

w = u + iv.

So u = ln(sqrt [x

^{2}+ y

^{2}] )

Question says u = constant lines get mapped in z plane as ?

u = constant

so ln (sqrt (x

^{2}+ y

^{2 }) = constant.

Now how to proceed? In answer they've said concentric circles with radius e

^{c}.

I got till u = ln (sqrt (x

^{2}+ y

^{2})

remove sqrt out as 1/2 we get:

u = 1/2 ( ln (x

^{2}+ y

^{2}) )

equation of circle is x

^{2}+ y

^{2}= constant.

But there is log in u.