This is not a homework problem - it was in a review book that I am studying before finals. The problem stated: Verify the Cauchy-Riemann equations for the following functions. Deduce that they are analytic. The function I am having trouble with is [tex] f(z) = z e^{-z} [/tex](adsbygoogle = window.adsbygoogle || []).push({});

Now, before I show my work, I just want to say that it seems quite obvious that this function is analytic, without the use of the Cauchy-Riemann functions.

[tex] g(z) = z [/tex], the identity map, is entire.

[tex] h(z) = e^{z} [/tex] is also entire, and never equal to zero.

Therefore, since [tex] f(x) = ze^{-z} = \frac{g(z)}{h(z)} [/tex] is the quotient of two entire functions (and the denominator is never 0) it must also be entire.

I also checked that the Cauchy-Riemann equations work for both functions [tex] g [/tex] and [tex] \frac{1}{h} [/tex] individually, but when I multiply them together, it doesn't work out. Where's my mistake?

[tex] z = x + iy [/tex]

[tex] ze^{-z} = (x + iy)(e^{-x}(cos y - i sin y)) = x e^{-x} cos y + y sin y + i (y e^{-x} cos y - x sin y) [/tex]

So, let the real part of this function be [tex] U(x,y) [/tex] and let the imaginary part be [tex] V(x,y) [/tex].

The Cauchy-Riemann equations say that [tex] U_x = V_y [/tex] and [tex] U_y = - V_x [/tex]. But I get:

[tex] U_x = -x e^{-x} cos y + e^{-x} cos y [/tex]

[tex] V_y = -y e^{-x} sin y + e^{-x} cosy - x cos y [/tex]

Clearly, these aren't the same...

Thanks for your help!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Cauchy-Riemann Equations problem (f(z) = ze^-z)

**Physics Forums | Science Articles, Homework Help, Discussion**