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Conversion from the Cartesian form to the polar form | Conversion from the Cartesian form to the polar form | ||
:<math>r = \sqrt{x^2+y^2}</math> | :<math>r = \sqrt{x^2+y^2}</math> | ||
− | :<math>\varphi = \arg(z) = \operatorname{ | + | :<math>\varphi = \arg(z) = \operatorname{artan}(y,x)</math> |
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. | 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. |
Latest revision as of 21:20, 4 September 2008
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.
Conversion from the polar form to the Cartesian form
- $ x = r \cos \varphi $
- $ y = r \sin \varphi $
Conversion from the Cartesian form to the polar form
- $ r = \sqrt{x^2+y^2} $
- $ \varphi = \arg(z) = \operatorname{artan}(y,x) $
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.