Imaginary part

In mathematics, the imaginary part of a complex number $z$ , is the second element of the ordered pair of real numbers representing $z,$ i.e. if $z = (x, y)$ , or equivalently, $z = x + i y$ , then the imaginary part of $z$ is $y$ . It is denoted by $Im z$ or $\Im z$ . The complex function which maps $z$ to the imaginary part of $z$ is not holomorphic.

In terms of the complex conjugate $\bar{z}$ , the imaginary part of z is equal to $\frac{z-\bar{z}}{2\mathrm{i}}$ .

For a complex number in polar form, $z = (r,θ)$ , or equivalently, $z = r (c o s θ + i s i n θ)$ , it follows from Euler's formula that $z = r e iθ$ , and hence that the imaginary part of $r e iθ$ is $r sinθ$ .

Imaginary part

See also

See also: Imaginary part, Complex conjugate, Complex function, Complex number, Euler's formula, Holomorphic, Imaginary number, Mathematics, Polar coordinates, Real number