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arg

Denotes the argument (or angle) of a complex number in mathematical notation.

Overview

Essential in complex analysis and related mathematical fields for expressing the angular component of complex numbers in polar form.

  • Commonly used alongside absolute value notation to fully describe complex numbers
  • Appears frequently in signal processing and electrical engineering applications
  • Often encountered in problems involving complex roots and polar coordinates
  • Typically returns values in the interval (-π, π] when using the principal argument

Examples

Finding the argument of a complex number z.

arg⁡(z)=θ\arg(z) = \theta 
\arg(z) = \theta

Expressing the argument in Euler's formula.

eix=cos⁡x+isin⁡x=∣eix∣eiarg⁡(eix)e^{ix} = \cos x + i\sin x = |e^{ix}|e^{i\arg(e^{ix})} 
e^{ix} = \cos x + i\sin x = |e^{ix}|e^{i\arg(e^{ix})}

Showing the relationship between complex logarithm and argument.

log⁡(z)=ln⁡∣z∣+iarg⁡(z)\log(z) = \ln|z| + i\arg(z) 
\log(z) = \ln|z| + i\arg(z)