I'm currently studying Dirichlet and Neumann problems from the Schaum series on Complex analysis, book.
I came across this phrase in the book: It is of interest that a Neumann problem can be stated in terms of an appropriately stated Dirichlet problem
I'm not being able to understand this statement. How exactly do I prove this statement? I've been asked to find a harmonic function in the upper half of the complex plane, and its normal derivative has the value $C$ on the positive $x$ axis and $0$ on the negative $x$ axis.
If this had been a dirichlet problem, I could have easily solved it. However, how do I convert the above neumann problem into a dirichlet problem ?
Problem: Find a harmonic function $u(x,y)$ in the upper half of the complex plane, and its normal derivative has the value $C$ on the positive $x$ axis and $0$ on the negative $x$ axis.
We consider a Dirichlet problem :
Find a harmonic function $U(x,y)(=u_y(x,y))$ in the upper half of the complex plane satisfying $U(x,0)=C,$ $x>0$ and $U(x,0)=0, $ $x<0$. We can easily find $$ U(x,y)=C-\frac{C}{\pi}\arg z, \quad (\,z=x+iy\,). $$ Let $f(z)=-v+iu$, where $v$ is the conjugate of $u$, then $f(z)$ is holomorphic and $f^\prime(z)=-v_x+iu_x=u_y+iu_x=U+iu_x$. Since $\operatorname{Re} f^\prime(z)=U,$ we see $f^\prime(z)=C+\frac{iC}{\pi}\log z.$ Then $f(z)=Cz+\frac{iC}{\pi}(z\log z -z)+D.$ Since $u$ is the imaginary part of $f(z)$, we have $$ u(x,y)=Cy+\frac{C}{\pi}(x\log |z| -y\arg z -x).$$ We have neglected a constant.