Complex variable example

1. 1.Prove that, |z1+z2|2+|z1-z2|2=2|z1|2+|z2|2
Solution:
Let, z1 = x1+iy1 & z2 = x2+iy2
So, |z1|= √( x1
2
+y1
2) & |z2| = √(x2
2+y2
2)
Or, |z1|2
= ( x1
2
+y1
2) & |z2| = (x2
2+y2
2)……………….1
Again we have, z1+z2= x1+iy1 + x2+iy2
Or, z1+z2 = (x1+ x2)+ i(y1 +y2)
So,| z1+z2 |2=√{(x1+ x2)2+ (y1 +y2)2}………….2

2. Now,
z1-z2 = x1+iy1 - x2-iy2
Or, z1-z2 = (x1-x2)+ i(y1 -y2)
So, | z1+z2 |2=√{(x1-x2)2+ (y1 -y2)2}………….3

3. Adding (2) & (3) , we get
| z1+z2 |2+| z1-z2 |2=(x1+ x2)2+ (y1 +y2)2+(x1-x2)2+ (y1 -y2)2
Or,| z1+z2 |2+| z1-z2 |2=(x1+ x2)2+(x1-x2)2+ (y1 +y2)2+ (y1 -y2)2
Or,| z1+z2 |2+| z1-z2 |2=2(x1
2+ x2
2)+ 2(y1
2+y2
2)
Or,| z1+z2 |2+| z1-z2 |2=2(x1
2+ y1
2)+ 2(x2
2+y2
2)
So, | z1+z2 |2+| z1-z2 |2=2|z1|2+|z2|2
proved