Argument (complex analysis)

 ( Trigonometry ) 

In some sources the argument is defined as $\backslash operatorname\{Arg\}(x\; +\; iy)\; =\; \backslash arctan(y/x),$ however this is correct only when , where $y/x$ is well-defined and the angle lies between $-\backslash tfrac\backslash pi2$ and $\backslash tfrac\backslash pi2.$ Extending this definition to cases where is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the half-plane and the two quadrants with , and then patch the definitions together:

$\backslash operatorname\{Arg\}(x\; +\; iy)\; =\; \backslash operatorname\{atan2\}(y,\backslash ,\; x)\; =$

 \arctan\left(\frac y x\right)       &\text{if } x > 0, \\[5mu]
 \arctan\left(\frac y x\right) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\[5mu]
 \arctan\left(\frac y x\right) - \pi &\text{if } x < 0 \text{ and } y < 0, \\[5mu]
 +\frac{\pi}{2}   &\text{if } x = 0 \text{ and } y > 0, \\[5mu]
 -\frac{\pi}{2}   &\text{if } x = 0 \text{ and } y < 0, \\[5mu]
 \text{undefined} &\text{if } x = 0 \text{ and } y = 0.
     

See atan2 for further detail and alternative implementations.

Arg[x + y I] $=\; \backslash begin\{cases\}$

 \text{undefined} &\text{if } |x| = \infty \text{ and } |y|=\infty, \\[5mu]
 0 &\text{if } x = 0 \text{ and } y = 0, \\[5mu]
 0 &\text{if } x = \infty, \\[5mu]
 \pi &\text{if } x = -\infty, \\[5mu]
 \pm\frac{\pi}{2} &\text{if } y = \pm\infty, \\[5mu]
 \operatorname{Arg}(x + y i) &\text{otherwise}.
     

or using the language's ArcTan:

Arg[x + y I] $=\; \backslash begin\{cases\}$

 0 &\text{if } x = 0 \text{ and } y = 0, \\[5mu]
 \text{ArcTan[x, y]} &\text{otherwise}.
     

ArcTan[x, y] is $\backslash operatorname\{atan2\}(y,\; x)$ extended to work with infinities. ArcTan[0, 0] is Indeterminate (i.e. it's still defined), while ArcTan[Infinity, -Infinity] doesn't return anything (i.e. it's undefined).

$\backslash begin\{cases\}$

 \frac{1\pi}{4} &\text{if } x = \infty \text{ and } y = \infty, \\[5mu]
 -\frac{1\pi}{4} &\text{if } x = \infty \text{ and } y = -\infty, \\[5mu]
 \frac{3\pi}{4} &\text{if } x = -\infty \text{ and } y = \infty, \\[5mu]
 -\frac{3\pi}{4} &\text{if } x = -\infty \text{ and } y = -\infty.
     

Unlike in Maple and Wolfram language, MATLAB's atan2(y, x) is equivalent to angle(x + y*1i). That is, atan2(0, 0) is $0$.

$z\; =\; \backslash left|\; z\; \backslash right|\; e^\{i\; \backslash operatorname\{Arg\}\; z\}.$

This is only really valid if is non-zero, but can be considered valid for if is considered as an indeterminate form—rather than as being undefined.

Some further identities follow. If and are two non-zero complex numbers, then

$\backslash begin\{align\}$

                   \operatorname{Arg}(z_1  z_2) &\equiv \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \pmod{\mathbb{R}/2\pi\mathbb{Z}}, \\
 \operatorname{Arg}\left(\frac{z_1}{z_2}\right) &\equiv \operatorname{Arg}(z_1) - \operatorname{Arg}(z_2) \pmod{\mathbb{R}/2\pi\mathbb{Z}}.
     

If and is any integer, then

$\backslash operatorname\{Arg\}\backslash left(z^n\backslash right)\; \backslash equiv\; n\; \backslash operatorname\{Arg\}(z)\; \backslash pmod\{\backslash mathbb\{R\}/2\backslash pi\backslash mathbb\{Z\}\}.$

$\backslash operatorname\{Arg\}\backslash biggl(\backslash frac\{-1-\; i\}\{i\}\backslash biggr)\; =\; \backslash operatorname\{Arg\}(-1\; -\; i)\; -\; \backslash operatorname\{Arg\}(i)\; =\; -\backslash frac\{3\backslash pi\}\{4\}\; -\; \backslash frac\{\backslash pi\}\{2\}\; =\; -\backslash frac\{5\backslash pi\}\{4\}$

• Lars Ahlfors (1979). 9780070006577, McGraw-Hill. ISBN 9780070006577
• S. Ponnuswamy (2025). 9788173196294, Narosa. ISBN 9788173196294
• Alan Beardon (1979). 9780471996712, Wiley. ISBN 9780471996712
• (2025). 000710295X, HarperCollins. 000710295X

• Argument at Encyclopedia of Mathematics.