Complex Variables: Prove f is Constant

Therefore exp(f(z)) must be a constant function. In summary, by considering the hint and using Liouville's theorem, we can show that f is constant.
  • #1
CornMuffin
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5

Homework Statement



Let z be a complex variable

Suppose f is an entire function and [tex]Re(f(z))\leq c[/tex] for all z

Show that f is constant.
(Hint: Consider exp(f(z))

Homework Equations


possibly this: [tex]e^z=e^x(cos(y)+isin(y))[/tex] where [tex]z=x+iy[/tex]

The Attempt at a Solution


I had no idea how I would show this, so I just started off trying a few things:
I first started off working with the hint to consider exp(f(z)), where exp((f(z))=ef(z)
I set g(z) equal to exp((f(z)) and because f(z) is entire, g(z) would also have to be entire
I first found a formula for the derivative of g(z) but that got me nowhere

I also tried working off the fact that [tex]Re(g(z))\leq e^ccos(Im(f(z)))[/tex]
but that got me nowhere as well...

I have been thinking about this problem for so long now, and I couldn't think of a way to show that f is constant
 
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  • #2
Do you know Liouville's theorem? |exp(f(z))|<=exp(c).
 

FAQ: Complex Variables: Prove f is Constant

1. What are complex variables?

Complex variables are mathematical functions that involve both real and imaginary numbers. They are commonly used in the field of mathematics to study and analyze systems that have both real and imaginary components.

2. How do you prove that a complex function is constant?

In order to prove that a complex function, f, is constant, you must show that f(z) is equal to a constant, C, for all values of z in the complex plane. This can be done by using techniques such as the Cauchy-Riemann equations and the Cauchy integral theorem.

3. What is the Cauchy-Riemann equation?

The Cauchy-Riemann equation is a set of necessary and sufficient conditions for a complex function to be differentiable at a point. It states that if a complex function, f(z), is differentiable at a point z = x + iy, then the partial derivatives of its real and imaginary components, u(x,y) and v(x,y), must satisfy the equations ux = vy and uy = -vx.

4. What is the Cauchy integral theorem?

The Cauchy integral theorem states that the integral of a complex function, f, along a closed contour is equal to 0 if f is analytic (differentiable) at all points within the contour. This theorem is useful in proving that a complex function is constant, as a constant function is always differentiable at all points within a contour.

5. Why is it important to prove that a complex function is constant?

Proving that a complex function is constant is important because it allows us to simplify and better understand complex systems. It is also a fundamental concept in complex analysis, which has many applications in fields such as physics, engineering, and economics.

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