stihl29
Jan19-10, 09:26 PM
1. The problem statement, all variables and given/known data
Suppose we already know series u(z) = \displaystyle\sum_{n=0}^\infty u_n(z)is uniformly convergent in the entire complex plain and we can perform term by term integration and differentation each term u_n(z) in the analyitic function. use cauchy-riemann equations to show that the sum of u(z) is analytic in the entire complex plain.
2. Relevant equations
$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},$
3. The attempt at a solution
My only guess at a solution would be to use the CR equations for each term in the sequence. ??
Suppose we already know series u(z) = \displaystyle\sum_{n=0}^\infty u_n(z)is uniformly convergent in the entire complex plain and we can perform term by term integration and differentation each term u_n(z) in the analyitic function. use cauchy-riemann equations to show that the sum of u(z) is analytic in the entire complex plain.
2. Relevant equations
$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},$
3. The attempt at a solution
My only guess at a solution would be to use the CR equations for each term in the sequence. ??