CornMuffin
- 51
- 5
Homework Statement
Let z be a complex variable
Suppose f is an entire function and Re(f(z))\leq c for all z
Show that f is constant.
(Hint: Consider exp(f(z))
Homework Equations
possibly this: e^z=e^x(cos(y)+isin(y)) where z=x+iy
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 Re(g(z))\leq e^ccos(Im(f(z)))
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