Consider the function defined by . Show that satisfies the Cauchy-Riemann equations at the origin yet it is not holomorphic at zero.

I can show that it satisfies the C-R equations but unsure about showing it is not holomorphic at zero. Please help?

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- Apr 25th 2013, 04:25 AMCuriosityCabinetCauchy-Riemann equations and holomorphic functions
Consider the function defined by . Show that satisfies the Cauchy-Riemann equations at the origin yet it is not holomorphic at zero.

I can show that it satisfies the C-R equations but unsure about showing it is not holomorphic at zero. Please help? - Apr 25th 2013, 05:43 AMxxp9Re: Cauchy-Riemann equations and holomorphic functions
f(z)-f(0)/(z-0) = f(z)/z

let z goes to 0 from the positive x axis, we have f(z)/z = 0/z = 0

let z goes to 0 from the positive part of the line y=x, we have f(z)/z = x/(x+ix)=(1-i)/2 - Apr 25th 2013, 05:51 AMCuriosityCabinetRe: Cauchy-Riemann equations and holomorphic functions
- Apr 25th 2013, 06:44 AMHallsofIvyRe: Cauchy-Riemann equations and holomorphic functions
A function, of a complex variable, is "holomorphic" at a point if and only if it is differentable there. What xxp9 did is show that the limit defining the derivative does not exist by showing that the limits, as z approaches 0 from two different directions, give different resuts