Page 126 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

116 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
where u and v are real functions of x and y. The derivative of a complex variable is defined as
follows:

f (z + z) − f (z)

f = lim
z (7.17)
z → 0
or

u(x + x, y + y) + iv(x + x, y + y) − u(x, y) − iv(x, y)

f (z) = lim
x + i y
x, y → 0 (7.18)

Taking the limit on x first, we find that

u(x, y + y) + iv(x, y + y) − u(x, y) − iv(x, y)
f (z) = lim

i y (7.19)
y → 0
and now taking the limit on y,

1 ∂u ∂v ∂v ∂u

f (z) = + = − i (7.20)
i ∂y ∂y ∂y ∂y
Conversely, taking the limit on y first,

u(x + x, y) + iv(x + x, y) − u(x, y) − iv(x, y)

j (z) = lim
x
x → 0 (7.21)
∂u ∂v
= + i
∂x ∂x
The derivative exists only if

∂u ∂v ∂u ∂v
= and =− (7.22)
∂x ∂y ∂y ∂x
These are called the Cauchy—Riemann conditions, and in this case the function is said to
be analytic. If a function is analytic for all x and y it is entire.
Polynomials are entire as are trigonometric and hyperbolic functions and exponential
functions. We note in passing that analytic functions share the property that both real and
2
2
imaginary parts satisfy the equation ∇ u =∇ v = 0 in two-dimensional space. It should be
obvious at this point that this is important in the solution of the steady-state diffusion equation