Page 36 - Calculus with Complex Numbers

J w éh u éh t? éh u éh t?
= + i = -ï + .
d c /7 /7 'i- 'J-y
y
Therefore on equating real and imaginal'y parts we have

bu gr bv gu
91 by 91 by

These are the Cauchy-Riemann equations published independently by Cauchy
(1818) and Riemann (1851).
W e call the formula

J w éh u éh t?
= - + i -
dz 9.:7 9.:7

the Cauchy-Riemann formula for the derivative.
In the case w = .:2 we get

éh u éh t? 9 t? bu
- = - = tz.x , - = - - = 2y.
9.:7 éh y 9.1 èy
Also the Cauchy-lkiemann formula gives

J w éh u éh t?
= - + i - = 2.x + liy = 2.
z
dz 9.:7 9.:7
as expected.

3.3 Failure of the C auchy-Riem ann equations
Corlsider the function w = @ = .x - iy. lf w = u + iv then we have u = .x
t? = -y. Therefore

bu 9 r éh t? éh u
= 1, = - 1, = - = (),
91 9 y 9.:7 èy

which means that the lirst Cauchy-lkiemann equation is not satislied for any .x , y.
We are forced to the conclusion that the function flz) = i cannot be differentiable
for any z.
ln this connection we have the following theorem .