(New page: 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 m...) |
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[edit] Conversion from the polar form to the Cartesian form | [edit] Conversion from the polar form to the Cartesian form | ||
| − | + | :<math> x = r\,(\cos \varphi)\,</math> | |
| − | :<math> y = r\,(\ | + | :<math> y = r\,(\sin \varphi )\,</math> |
[edit] Conversion from the Cartesian form to the polar form | [edit] Conversion from the Cartesian form to the polar form | ||
| − | + | :<math> r = \sqrt(\frac{x^2}+\frac{y^2}),</math> | |
| − | + | \varphi = arg (z) =atan2(y,x) | |
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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. | ||
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Revision as of 17:14, 3 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.
[edit] Conversion from the polar form to the Cartesian form
- $ x = r\,(\cos \varphi)\, $
- $ y = r\,(\sin \varphi )\, $
[edit] Conversion from the Cartesian form to the polar form
- $ r = \sqrt(\frac{x^2}+\frac{y^2}), $
\varphi = arg (z) =atan2(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.
