Conformal Solution

By letting $w\equiv f(z)$, the Real and Imaginary Parts of $w$ must satisfy the Cauchy-Riemann Equations and Laplace's Equation, so they automatically provide a scalar Potential and a so-called stream function. If a physical problem can be found for which the solution is valid, we obtain a solution--which may have been very difficult to obtain directly--by working backwards. Let

$$Az^n = Ar^ne^{in\theta},$$ (1)

the Real and Imaginary Parts then give

$$\phi = Ar^n\cos(n\theta)$$ (2)

$$\psi = Ar^n\sin(n\theta).$$ (3)

For $n=-2$,

$$\phi = {A\over r^2}\cos(2\theta)$$ (4)

$$\psi = -{A\over r^2}\sin(2\theta),$$ (5)

which is a double system of Lemniscates (Lamb 1945, p. 69). For $n=-1$,

$$\phi = {A\over r}\cos\theta$$ (6)

$$\psi = -{A\over r}\sin\theta.$$ (7)

This solution consists of two systems of Circles, and $\phi$ is the Potential Function for two Parallel opposite charged line charges (Feynman et al. 1989, §7-5; Lamb 1945, p. 69). For $n=1/2$,

$$\phi = Ar^{1/2}\cos\left({\theta\over 2}\right)= A\sqrt{\sqrt{x^2+y^2}+x\over 2}$$ (8)

$$\psi = Ar^{1/2}\sin\left({\theta\over 2}\right)= A\sqrt{\sqrt{x^2+y^2}-x\over 2}\,.$$ (9)

$\phi$ gives the field near the edge of a thin plate (Feynman et al. 1989, §7-5). For $n=1$,

$$\phi = Ar\cos\theta=Ax$$ (10)

$$\psi = Ar\sin\theta=Ay.$$ (11)

This is two straight lines (Lamb 1945, p. 68). For $n=3/2$,

$$w=Ar^{3/2}e^{3i\theta/2}.$$ (12)

$\phi$ gives the field near the outside of a rectangular corner (Feynman et al. 1989, §7-5). For $n=2$,

$$w=A(x+iy)^2=A[(x^2-y^2)+2ixy]$$ (13)

$$\phi = A(x^2-y^2)=Ar^2\cos(2\theta)$$ (14)

$$\psi = 2Axy=Ar^2\sin(2\theta).$$ (15)

These are two Perpendicular Hyperbolas, and $\phi$ is the Potential Function near the middle of two point charges or the field on the opening side of a charged Right Angle conductor (Feynman 1989, §7-3).

See also Cauchy-Riemann Equations, Conformal Map, Laplace's Equation

References

Feynman, R. P.; Leighton, R. B.; and Sands, M. The Feynman Lectures on Physics, Vol. 1. Redwood City, CA: Addison-Wesley, 1989.

Lamb, H. Hydrodynamics, 6th ed. New York: Dover, 1945.