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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]

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!

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!

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