Question

Mean value property harmonic functions in complex analysis proof

Answer 1

Answer:

Step-by-step explanation:

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R where U is an open subset of Rn that satisfies Laplace's equation, that is,

{\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0}  \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0

everywhere on U. This is usually written as

{\displaystyle \nabla ^{2}f=0}  \nabla^2 f = 0  

or

{\displaystyle \textstyle \Delta f=0} \textstyle \Delta f = 0