This question involved complex numbers?

Question

Use Cauchy-riemann equation to find if the complex function (f (z) = z sq - iz) is analytic in the complex plane. Show the real and imaginary parts of f is harmonic

Answer 1

Given a complex variable
w = u(x,y) + iv(x,y) = f(z) the Cauchy-Riemann requirements for analyticity (continuity of the function and it's derivatives) at some point (say a + ib) requires
(partial u)/(partial x) = (partial v)/(partial y) and
(partial v)/(partial x) = -(partial u)/(partial y)

I know you've had enough Math to be able to do partial derivatives, or you wouldn't even be asking this kind of question, so you do the arithmetic.

A function (say f(z)) is said to be 'harmonic' in a region R if for all z contained in R the sum of the 2'nd partials of f withrespect ot x and y is 0. So, again, go do the arithmetic for the function you have ☺

BTW..... The thing with analyticity and harmonic functions will get to be *real* important when you start doing things such as Greens Theorem and Stokes Theorem and get into vector Calculus, so it makes good sense to get to know them well at this point.


Doug

Answer 2

(5 + i)x + 2i = 6 x = (6 - 2i)/(5 + i) by potential of convention we don't bypass away the respond like that. We set up for it to have a actual denominator by potential of multiplying by potential of a million = (5 - i)/ (5 - i) (distinction of two squares identity) Do you undergo in strategies that bit now ? x = [(6 - 2i)(5 - i)][(5 + i)(5 - i)] x = [(30 +(-2)) - 10i - 6i]/[25 - (-a million)] x = [28 - 16i]/[26] x = 14/13 - (8/13)i wish that helped, Regards - Ian