ABSTRACT

Complex Variables If z = x + iy, where x and y are real variables, z is a complex variable and is a function of x, and y, z may be represented graphically in an Argand diagram (Figure 19.3).

Polar form:

z = x + iy = |z|eiθ = |z|(cosθ + isinθ) x = rcosθ

y = rsinθ

where r = |z|. Complex arithmetic:

z1 = x1 + iy1

z2 = x2 + iy2

z1 ± z2 = (x1 ± x2) + i(y1 ± y2)

z1 ·z2 = (x1x2 − y1y2) + i(x1y2 − x2y1)

Conjugate:

z* = x − iy

z ·z* = x2 + y2 = |z|2

Function: another complex variable w = u + iv may be related functionally to z by:

w = u + iv = f(x + iy) = f(z)

which implies:

u = u(x, y)

v = v(x, y)

For example:

coshz = cosh(x + iy) = coshxcosh iy + sinhxsinh iy = coshxcosy + isinhxsiny

u = coshxcosy

v = sinhxsiny

Cauchy-Riemann Equations If u(x, y) and v(x, y) are continuously differentiable with respect to x and y:

w = f(z) is continuously differentiable with respect to z and its derivative is:

It is also easy to show that ∇2u = ∇2v = 0. Since the transformation from z to w is conformal, the curves u = constant and v = constant intersect each other at right angles, so that one set may be used as equipotentials and the other as ﬁeld lines in a vector ﬁeld.