1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 23

Theorem 1.5 (Lagrange's Identity). If {z 1 , z 2 , ... , Zn} and {w1, w2, ... ,

wn} are two sets of n complex numbers (n 2:: 1), then

n^2 n n

Lzkwk = L lzkl

2

L lwkl

2

• L lz;wk -Zkw;l

(^2) (1.6-11)

k=l k=l · k=l l:Si9:Sn

Proof We have

I

t ZkWkl

2

= (z1w1 + · · · + ZnWn)(z1w1 + · · · + ZnWn)

k=l

= lz11^2 Jw11^2 + · · · + lznl^2 lwnl^2 + LZiWiZjWj

i#j

and

n n

= Jz1J

2

lw11

2

• · · · + lznl

2

lwnl

2

+ L(z;wiz;w; + z;wiz;w;)

i<j

L lzkl^2 L lwkl^2 = (iz11^2 + · · · + lznl^2 )(lw11

2

• · · · + lwnl

2

)

k=l k=l.

(1.6-12)

= Jz1l^2 iwiJ2 + · · · + lznl

2

lwnl

2

+ L Jz;J

2

1wil

2

i-:f.j

= lz1l^2 lw1l^2 + · · · + lznl

2

lwnl

2

+ L(z;.Z;WjWj + Zj.Z;w;w;)

i<j

Subtracting (1.6-12) from (1.6-13), we get

t, iz.1' t, 1w,1' -It, z,w, '

= L[z;wi(z;w; - z;w;) - z;w;(z;w; -ziw;)]

i<j

= L(z;wi - z;w;)(z;w; - z;w;)

i<j

= L jz;Wj - Zjw;J^2

i<i

which is equivalent to (1.6-11).

(1.6-13)

(1.6-14)