Page 37 - Calculus with Complex Numbers

Theorem 2 For u + i t? = f(x + iy) with continuous partial derivatives bu/bx
élu/él y, é) v/bx, g??/i y the function flz) is differentiable at z if and only if the
Cauchy-lkiemann equations

bu gr bv gu
91 by 91 by

are satislied.
Proof We proved necessity above. For sufliciency we refer the reader to rigorous
books on complex analysis.

3.4 G eom etri c signifi cance of the com p Iex derivative

For a real function f @) of a real variable .x the equation of the tangent to the
graph y = f @) at .x = a is

)' = f (tz) + (x - tz) f' (J).

For a complex function f (z) of a complex variable z, the equation of the tangent
plane (in 4 dimensions) to the graph w = flz) at z = a is

w = fla) + (z - a)f'(a) = ztz + B,

where .4 = f (tz), B = fla) - af (tz).
The geometric effect of the linear function w = ztz + B is a rotatioa a scaling,
and a translation. The rotation is through the angle arg .4 the scaling is by the

factor I.4 I. The translation is through a distance IB I in the direction arg B.

What this tells us about the transformation w = f (z) is that near z = a the
effect is approximately a rotation through arg f' (tz), and a scaling by I f (tz) I . For
example, if a = in'/l and flz) = ez, then we have fla) = ein'/l = i. Also
f' (tz) = cl = ein'/l = i . So the effect near z = a is a rotation through 90O
anticlockwise (see Figure 3.4). lf b = in'/l + 1, then we have flb) = f'Lb) =
ei, so the effect locally is now a scaling by c, and again a rotation through 90O
anticlockwise (see Figtlre 3.4).
The fact that w = flz) acts locally like a rotation through arg f (z) explains
why curves which intersect at a certain angle in the z-plane are transformed under
the action of w = flz) to curves which intersect at the same angle in the w-plane.
This is the characteristic property of a conformal flll////ïrl,j' which is important for
the applications to lluid mechanics.