Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number.
[edit] Conversion from the polar form to the Cartesian form
:$ x = r\,(\cos \varphi)\, $
- $ y = r\,(\cos \varphi )\, $
[edit] Conversion from the Cartesian form to the polar form
(See arg function and atan2.)
The resulting value for φ is in the range (−π, +π]; it is negative for negative values of y. If instead non-negative values in the range [0, 2π) are desired, add 2π to negative results.
[edit] Notation of the polar form
The notation of the polar form as
is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as
which is called exponential form.
