1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

Then we have

and

8zf = fz =^1 Mfx - ify)

az1=fz=^1 /2Ux + ify)

fx = fz + Jz, fy = i(fz - fz)

With this new notation, equation (6.7-5) becomes

df = fzdz + f -zdz

We note that

fz = %[(ux + ivx)-i(uy +ivy)]

= %[(ux + vy) + i(vx -Uy)]

fz =^1 / 2 [(ux + ivx) + i(uy +ivy)]

=^1 / 2 [(ux - Vy)+ i(vx + uy)]

From (6.7-8) and (6.7-9) it follows that

and

'lfzl^2 =^1 / 4 [(ux + vy)^2 + (vx -.uy)^2 ]

lfzl^2 =^1 /4[(ux - vy)^2 + (vx + uy)^2 ]

327

(6. 7-7)

(6.7-8)

(6.7-9)

(6.7-10)

By addition and subtraction of equations (6.7-8) and (6.7-9) we get

f z + f z = Ux + ivx

i(fz - fz) =Uy +ivy

(6.7-11)

(6.7-12)

We make use of (6.7-11) and (6.7-12) in Section 6.10. If f is monogenic at

z, then Ux = Vy and Uy = -Vx and (6.7-9) becomes

fz = 0 (6.7-13)

Equation (6.7-13) is to be considered as the complex form of the Cauchy-
Riemann equations.
Let P be a function (or operator) acting on a set S. Then the set

K ={a: a E S,Pa = O} is called the kernel of P .. Equation (6.7-13) shows

that the kernel of the operator 8-z acting on 'D(A) is the family H(A) of
analytic functions in A.
Also, if the equations Ux =Vy, Uy = -vx are satisfied, equation (6.7-8)
becomes

fz = Ux + ivx = f'(z) (6.7-14)