Mathematics of Physics and Engineering

192 Functions of a Complex Variable

e(z) = E(x,y) + ie2(x,y), f'(zo) = A + iB, r = \z\, we find that, for
all x, y sufficiently close to zero,

(u(x 0 + x,y 0 + y) -u(x 0 ,yo)) + i(v(x 0 + x,y 0 + y) -v(x 0 ,y 0 ))

= {x + iy)(A + IB) + ei(x,y) + ie 2 (x,y),

where \ek(x,y)/r\ —> 0, r —* 0, for k = 1,2.

EXERCISE 4.2.7. c By comparing the real and imaginary parts in the
last equality, convince yourself that the functions u, v are difjerentiahle at
{xo,yo) and the following equalities hold:

A = —{x 0 ,y 0 ) = —(x 0 ,y 0 ), B = - — (x 0 ,y 0 ) = ~(x 0 ,y 0 ).

We now state the main result of this section.

Theorem 4.2.1 A function f = f(z) is difjerentiahle at the point ZQ =
^o + iya if and only if the functions u = 3?/, v = S/ are difjerentiahle at
the point (xo,yo) and the equalities

dx dy' dy dx

hold at the point (xo,yo).

Equalities (4.2.2) are known as the Cauchy-Riemann equations. Riemann
introduce many key concepts of complex analysis in 1851 in his Ph.D. dis-
sertation; his advisor was Gauss. And, as we mentioned earlier in connec-
tion with the Cauchy-Bunyakovky-Schwartz inequality, one should not be
surprised to see the name of Cauchy attached to an important result.

EXERCISE 4.2.8. (a)B Prove the above theorem. Hint: You already have
the proof in one direction, and you reverse the arguments to get the other direc-
tion. (b)c Verify that the derivative f'(z) can be written in one of the four
equivalent ways:
,.. du .dv dv .du du .du dv .dv
dx dx dy dy dx dy dy dx'

(c)B Verify that if the function f is analytic in the domain G, then each
of the following implies that f is constant in G: (i) f'(z) = 0 in G; (ii)
Either 3?/ or 9/ is constant in G; (Hi) \f(z)\ is constant in G.

We will see later that if a function / = f(z) is differentiable (analytic)
in a domain, then it is infinitely differentiable there. Then the functions u