- #1

- 55

- 0

## Homework Statement

Let gama be a closed curve and f be analytic function. Show that the integration of f(z)f' dz is puerly imaginary

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter sbashrawi
- Start date

- #1

- 55

- 0

Let gama be a closed curve and f be analytic function. Show that the integration of f(z)f' dz is puerly imaginary

- #2

- 336

- 0

As f(z)f'(z) = (1/2) d/dz f^2 , Cauchy's formula shows that what you claim is invalid unless gamma encircles some poles of f with real residues at them.

- #3

- 55

- 0

I am sorry the true statment is that :

integration of ( conjugate of f ) * f' *dz is purely imaginary.

I tried to prove it using the winding number but I couldn't

- #4

- 1,838

- 7

Take the real part of the integral expression by adding the complex conjugate.

- #5

- 55

- 0

here is what I did:

integ(conj(f)*f'dz) = integr( f + conj(f))*f'dz

which implies

integ [ conj(f) - 2 Re(f)] * f' dz = 0

letting f = u + iv , then the expression will be

integ[ -u -iv] * f' dz = 0

then I couldn't find how it is purely imaginary from this step

- #6

- 1,838

- 7

integr( f + conj(f))*f'dz =

integr (2Re(f))*f'dz =

integr[(2 u) (du + i dv)] =

2i integr u dv

integr (2Re(f))*f'dz =

integr[(2 u) (du + i dv)] =

2i integr u dv

- #7

- 55

- 0

Thank you very much

Share: