Some basics of complex analysis. We will follow L. Ahlfors's Complex Analysis book. An analytic function is a complex-valued function with derivatives everywhere where function f(x) is defined with an appropriate power series. Holomorphic functions are the same, with a different meaning. For an analytic function f(z) = u+iv, we write
\frac{\partial f}{\partial x} = \frac{\partial u}{\partial x}+i\frac{\partial
v}{\partial x}
with the limit
f^{\prime}(z)=\lim _{k \rightarrow 0} \frac{f(z+i k)-f(z)}{i k}=-i
\frac{\partial f}{\partial y}=-i \frac{\partial u}{\partial y}+\frac{\partial
v}{\partial y}
from which we can extract
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad
\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.
These differential equations have a name; Cauchy-Reimann differential
equations. (They have well-meaning in complex analysis, greatly in Reimann
spheres, from my readings.) If one compute |f(z) |^2, which is a
Jacobian of u with respect to x and v respect to y, one can extract
from the Jacobian that \Delta u and \Delta v are harmonic functions as
\Delta u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2}
u}{\partial y^{2}}=0
\Delta v=\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2}
v}{\partial y^{2}}=0
and u and v satisfy the Cauchy-Reimann differential condition. v is
said to be a harmonic conjugate of u, u is a harmonic conjugate
of -v.
If u(x, y) and v(x, y) have continuous first-order partial derivatives
which satisfy the Cauchy-Riemann differential equations, then f(z)=u(z)+i
v(z) is analytic with continuous derivative f^{\prime}(z), and
conversely.
Edit: If one wants to go at an advanced level (which sure one needs in theoretical computations), try Stephen Fisher's Complex Variables, as suggested by one friend.
We will follow L. Alhfor's Complex Analysis book
-----------------
Lars V. Ahlfors.
Anonymous,
I should have checked the typos. Thank you.
I have always struggled with complex analysis but since I have done a lot of physics I find complex analyis inbuilt in physics
I did take a course in undegrad