$($a$)$ A two$-$dimensional source of strength $m,$ located at the origin, is such that the flow is axisymmetric and everywhere radial, and the total volume flux outwards from the origin is $m.$ Derive the complex potential for this source from first principles.

What is the complex potential for a source, strength $m,$ situated at a point $z_0?$

$($b$)$ Now consider a uniform flow of speed $U=2\frac{m}{}$a parallel to the $x-$axis, into which a source of strength $m$ is placed at $(-2a,0)$ and a sink of equal strength is placed at $(2a,0).$

Write down the complex potential for this configuration, and show that there are stagnation points at $z=+-a\sqrt[\phantom{2}]{5}.$

Find a general expression for the streamlines, and show that the streamline which has $y=0$ as one branch can be written as

$(x^2+y^2-4a^2)tan(\frac{4y}{a})=4ay.$

$[$Hints:

i$)$ You are given that, for a complex quantity $A,ln(A)=ln(|A|)+$iarg$(A).$

ii$)$ You may find it useful to use the identity

$arctan-arctan-=arctan(\frac{-}{1+})$