1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 553

Theorem 8.12 Let f(z) = u(x, y) + iv(x, y) and suppose that the func-
tions u and v are of class cn+i in some circular neighborhood Nr(z) of the
point z = (x, y). Then for z + 6.z E Nr(z) we have
n 1
f(z + 6.z) = f(z) + L k! (fz6.z + fz6.z)(k) +Rn
k=l

(8.6-1)

where the symbolic power has the usual interpretation, and Rn = R 1 ,n +
iR2,n, Ri,n and R 2 ,n being the remainders in the corresponding Taylor
formulas for u and v (as functions of the real variables x and y). ·

Proof From Taylor formula for real functions of two real variables we have
1 1
u(x + 6.x, y + 6.y) = u(x, y) +du+ I d^2 u + · · · + I dnu + Ri,n (8.6-2)

2. n.
   1 1
   v(x + 6.x, y + 6.y) = v(x, y) + dv +
   21
   d^2 v + · · · + n! dnv + Rz,n (8.6-3)

where

dnu = (u.,6.x + uy6.y)<n),

and

R - l dn+lu ] R - l dn+lv ]

l,n - (n + 1)! x+9Ax ' Z,n - (n + l)! x+9^1 Ax

y+9Ay y+9^1 Ay

with 0 < e < 1, 0 < O' < 1. From (8.6-2) and (8.6-3) we obtain

J(z+D..z) = J(z)+(du+idv)+ ~(d^2 u+id^2 v)+···+ ~(dnu+idnv)+Rn

2. n.

Now, from

fz = %[(u., + vy) + i(v., - Uy)]

fz =^1 / 2 [(u., - vy) + i(v., + uy)]

J zz =^1 / 4 [( Uxx + 2vxy - Uyy) + i( Vxx - 2Uxy - Vyy )]

etc.

it can be easily verified that

du+ idv = f z6.z + f -z6.z

d^2 u + id^2 v = f zz6.z^2 + 2fz-z6.z6.z + fz--z6.z^2

= (f z6.z + f-z6.z)<^2 )

and in general, we have

dnu + idnv = (fz6.z + fz6.z)(n)

(8.6-4)