1550251515-Classical_Complex_Analysis__Gonzalez_

450 Chapter?

If we put

j k

Zjk = h( N ' N ), j, k = 0, 1, ... 'N

j j+l k k+l

Sjk = [ N , JV] x [ N , 1\T ], j, k = 0, ... , N - 1

then h(S,;k) C Dr(Zjk) CA, where Dr(Zjk) = {z: lz - Zjkl < r }. Next let

ri+l,k

P(j,k) = lzjk. f(z)dz, (j, k) E LN-1,N

rj,k+l

Q(j,k) = Jzjk f(z)dz, (j, k) E LN,N-1

All three points Zjk, Zj+l,k, Zj,k+l lie in the disk Dr(Zjk), so as a conse-

quence of Corollary 7.9, the integrals above are independent of the path

lying in Dr(Zjk) and joining Zjk to Zj+i,k• or to Zj,k+l·

For (j, k) E LN-1,N-1, we have

(D.xQ - D.yP)(j, k)

= [Q(j + 1, k) -Q(j, k)] - [P(j, k + 1) -P(j, k)]

{

rzj+l,k+l {Zj,k+1 rzj+l,k+l {Zj+l,k}

= Jz - Jz - Jz + Jz J(z) dz

Zj+l,k Zjk Zj,k+l Zjk

{

ri+l,k ri+l,k+l rj,k+l rjk }

= Jz + Jz + Jz + Jz J(z) dz= O (^7 .lO-ll)

Zjk Zj+l,k Zj+l,k+l Zj,k+l

by CoroHary 7.9 as applied to the quadrilateral contour with vertices Zjk,

Zj+i,k, Zj+i,k+l, Zj,k+l contained in the disk Dr(Zjk)·

On the other hand,

J PD.x + QD.y

8LNN

N-1 N-1 N-1 N-1

= L P(j,O) + L Q(N, k)-L P(j,N)-L: Q(O, k)

j=O k=O j=O k=O

N-1 lh((j+l)/N,O) N-1

1

h(l,(k+l)/N)

·- L. f(z)dz+ L f(z)dz

j=O h(3/N,O) k=O h(l,k/N)

N-1

1

h((j+l)/N,1) N-1 lh(O,(k+l)/N)

• L. J(z)dz - L f(z)dz
  j=O h(3/N,l) k=O h(O,k/N)