# Documentation

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# angle

## Syntax

```P = angle(Z) ```

## Description

`P = angle(Z)` returns the phase angles, in radians, for each element of complex array `Z`. The angles lie between ±π.

For complex `Z`, the magnitude `R` and phase angle `theta` are given by

```R = abs(Z) theta = angle(Z)```

and the statement

`Z = R.*exp(i*theta)`

converts back to the original complex `Z`.

## Examples

collapse all

Create a matrix of complex values and compute the phase angle of each element.

```Z = [1 - 1i 2 + 1i 3 - 1i 4 + 1i 1 + 2i 2 - 2i 3 + 2i 4 - 2i 1 - 3i 2 + 3i 3 - 3i 4 + 3i 1 + 4i 2 - 4i 3 + 4i 4 - 4i]; P = angle(Z)```
```P = 4×4 -0.7854 0.4636 -0.3218 0.2450 1.1071 -0.7854 0.5880 -0.4636 -1.2490 0.9828 -0.7854 0.6435 1.3258 -1.1071 0.9273 -0.7854 ```

## Algorithms

The `angle` function takes a complex number z = x + iy and calculates `atan2(y,x)` to find the angle formed in the xy-plane between the positive x-axis and a ray from the origin to the point (x,y). This phase angle is also the imaginary part of the complex logarithm, since

`$\begin{array}{l}z=r{e}^{i\theta }\\ \mathrm{log}\left(z\right)=\mathrm{log}\left(r\right)+i\theta \text{\hspace{0.17em}}.\end{array}$`