- #1

fatpotato

- Homework Statement
- Prove that ##f(z) = Re\big(\frac{\cos z}{\exp{z}}\big)## is harmonic on the whole complex plane.

- Relevant Equations
- Extension of usual functions to the complex plane

Definition of ##\cos(z)## and ##\exp(z)## where ##z \in \mathbb{C}##

Cauchy-Riemann equations

Laplace equation

Hello,

I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane.

This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend to believe I have to use the definitions to extract the real part of ##f(z)## and prove that it solves the Laplace equation on ##\mathbb{C}##.

However, I have the feeling there is a more elegant solution than going through the tedious computation needed, involving the expansion of the cosine in exponential form : $$ f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) = \frac{1}{2} Re\big( \frac{\exp{iz} + \exp{-iz}}{\exp{z}} \big)$$

Is there a subtlety I am not seeing? For example, I can see that the derivative of ##f(z)## exists, is always continuous and that the denominator is never zero, and I would like to think ##f## is analytic, therefore proving without further computation that its real part indeed solves the Laplace equation, but I fear using this shortcut without showing that the partial derivatives of ##f## exist, are continuous and satisfy the Cauchy-Riemann equations.

Is it proof enough?

Thank you

I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane.

This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend to believe I have to use the definitions to extract the real part of ##f(z)## and prove that it solves the Laplace equation on ##\mathbb{C}##.

However, I have the feeling there is a more elegant solution than going through the tedious computation needed, involving the expansion of the cosine in exponential form : $$ f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) = \frac{1}{2} Re\big( \frac{\exp{iz} + \exp{-iz}}{\exp{z}} \big)$$

Is there a subtlety I am not seeing? For example, I can see that the derivative of ##f(z)## exists, is always continuous and that the denominator is never zero, and I would like to think ##f## is analytic, therefore proving without further computation that its real part indeed solves the Laplace equation, but I fear using this shortcut without showing that the partial derivatives of ##f## exist, are continuous and satisfy the Cauchy-Riemann equations.

Is it proof enough?

Thank you