Tool for calculating the value of the argument of a complex number. The argument of a nonzero complex number \( z \) is the value (in radians) of the angle \( \theta \) between the abscissa of the complex plane and the line formed by \( (0;z) \).

Complex Number Argument - dCode

Tag(s) : Arithmetics, Geometry

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!

You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? *Contact-me*!

Sponsored ads

Tool for calculating the value of the argument of a complex number. The argument of a nonzero complex number \( z \) is the value (in radians) of the angle \( \theta \) between the abscissa of the complex plane and the line formed by \( (0;z) \).

The argument is an angle \( \theta \) qualifying the complex number \( z \):

$$ \arg(z) = 2\arctan \left(\frac{\Im(z)}{\Re(z) + |z|} \right) = \theta \mod 2\pi $$

with \( \Re(z) \) the real part and \( \Im(z) \) the imaginary part of \( z \).

Example: Take \( z = 1+i \), the real part is \( 1 \), the imaginary part is \( 1 \) and the modulus of the complex number \( |z| \) equals \( \sqrt(2) \), so \( \arg(z) = 2 \arctan \left( \frac{1}{1 + \sqrt(2) } \right) = \frac{\pi}{4} \)

The result of the \( \arg(z) \) calculation is a value between \( -\pi \) and \( +\pi \) and the theta value is modulo \( 2\pi \)

The argument of \( 0 \) is \( 0 \).

Take \( z \), \( z_1 \) and \( z_2 \) be non-zero complex numbers and \( n \) is a natural integer. The remarkable properties of the argument function are:

$$ \arg( z_1 \times z_2 ) \equiv \arg(z_1) + \arg(z_2) \mod 2\pi $$

$$ \arg( z^n ) \equiv n \times \arg(z) \mod 2\pi $$

$$ \arg( \frac{1}{z} ) \equiv -\arg(z) \mod 2\pi $$

$$ \arg( \frac{z_1}{z_2} ) \equiv \arg(z_1) - \arg(z_2) \mod 2\pi $$

If \( a \) is a strictly positive real and \( b \) a strictly negative real, then

$$ \arg(a \cdot z) \equiv \arg(z) \mod 2\pi $$

$$ \arg(b \cdot z) \equiv \arg(z) +\pi \mod 2\pi $$

dCode retains ownership of the source code of the script Complex Number Argument online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be given for free. To download the online Complex Number Argument script for offline use on PC, iPhone or Android, ask for price quote on contact page !

argument,complex,number,angle,phase,plane,theta

Source : https://www.dcode.fr/complex-number-argument

© 2019 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode

Feedback