1550251515-Classical_Complex_Analysis__Gonzalez_

326. Chapter^6

Note If in (6.6-10) we let z = zo we get

f(zo) = u(xo,Yo) + iv(xo,yo) = 2u(xo,Yo) + C

so that C = -u(x 0 , Yo)+iv(xo, y 0 ). Letting v(xo, Yo)= Co and substituting

C = -u(xo,Yo)+iC 0 in (6.6-10), we obtain

f(z) = 2u [ ~(z + zo), ;i (z - zo)] -u(xo,yo) + iCo (6.6-12)

where Co denotes a real constant.

Similarly, (6.6-11) can be written as

f(z) = 2u ( ~, ;J -u(O, 0) + iCo

6. 7 THE CONCEPT OF' THE DIFFERENTIAL OF A

COMPLEX FUNCTION. COMPLEX DIFFERENTIAL

OPERATORS

(6.6-13)

Definitions 6.5 Let f(z) = u(x, y) + iv(x, y), where u(x, y) and v(x, y)

are differentiable functions of x and y in some open set A. Then the partial

derivatives off with respect to x and y, and the total differential off, are

defined in A by the formulas

f x = Ux + ivx, f y = Uy +ivy

df = f x dx + f y dy

(6.7-1)

(6.7-2)

where dx = D.x and dy = D.y are arbitrary increments of x and y. If f is

monogenic at z, in view of (6.5-4) we have

fx = -ify (6.7-3)

which combines in one equation the Cauchy-Riemann equations (6.5-3).

On applying (6.7-2) to the particular functions z = x +iy and z = x _.:.iy,

we find

dz= dx + idy,

and it :follows that

1

dx =

2

(dz+ dz),

dz= dx - idy

i

dy = - -(dz - dz)

2

Substitution of (6.7-4) in (6.7-2) yields

df = ~ Ux - ify) dz+ ~ Ux + ify) dz

(6.7-4)

(6.7-5)

This suggests the introduction of the two complex differential operators

8=~(~-ii_) z 2 ax 8y ' (6.7-6)