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

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In summary, the problem is to verify the Cauchy-Riemann equations for the function f(z) = ze^(-z) and determine if it is analytic. The conversation discusses the steps taken to show that f(z) is indeed analytic, using the fact that it is the quotient of two entire functions. A mistake in the calculation is identified and corrected.
  • #1
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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!
 
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  • #2
The exponent [itex]e^{-x}[/itex] should be a common factor for all of your terms.. but in the line where you multiply everything out, you seem to have dropped it from some terms..

[tex] ze^{-z} = (x + iy)(e^{-x}(cos y - i sin y)) \neq x e^{-x} cos y + y sin y + i (y e^{-x} cos y - x sin y) [/tex]
 
  • #3
Oh, I see... duh.

Thanks!
 
  • #4
In your equation for ze-z, your expressions for U and V are incorrect. e-x multiplies all terms.
 

1. What are the Cauchy-Riemann equations?

The Cauchy-Riemann equations are a set of two partial differential equations that describe the behavior of complex-valued functions. They are used to determine if a function is analytic, meaning it can be represented by a power series, and to find the derivative of a complex function.

2. How do the Cauchy-Riemann equations relate to the function f(z) = ze^-z?

The Cauchy-Riemann equations can be used to determine if f(z) = ze^-z is an analytic function. If the partial derivatives of the real and imaginary parts of this function satisfy the Cauchy-Riemann equations, then the function is analytic and can be represented by a power series.

3. What is the significance of the Cauchy-Riemann equations in complex analysis?

The Cauchy-Riemann equations are fundamental in complex analysis. They provide a way to determine if a function is analytic and can be represented by a power series. They are also used to find the derivative of a complex function, which is essential in many areas of mathematics, physics, and engineering.

4. What are the applications of the Cauchy-Riemann equations?

The Cauchy-Riemann equations have many applications in mathematics and other fields. They are used in the study of complex functions, harmonic functions, conformal mapping, and potential theory. They are also used in physics to describe the behavior of electric and magnetic fields.

5. Are there any limitations to the Cauchy-Riemann equations?

While the Cauchy-Riemann equations are powerful tools in complex analysis, they do have some limitations. They can only be used to determine if a function is analytic at a single point, so they cannot be used to determine the analyticity of a function on an interval or region. Additionally, they cannot be used to find the derivative of a non-analytic function.

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