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Analytic Functions

15.0 PRELIMINARIES

A complex number z is of the form x + iy, where x and y are real numbers and i = √−1 is called the imaginary unit.

x is called the real part of z and is denoted as Re z.

y is called the imaginary part of z and is denoted as Im z.

Thus, x = Re z, y = Im z.

1. Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, written as z1 = z2, if and only if x1 = x2 and y1 = y2.

Note Given two complex numbers z1 and z2, we can only say z1 = z2 or z1z2. We cannot say z1 < z2 because there is no order relation in the field of complex numbers as in the field of real numbers.

The set of all complex numbers is denoted by C.

1. Complex conjugate

If z = x + iy is any complex number, then its conjugate

We can easily ...

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