Definition and Properties of Argument (Complex Analysis)
– The argument of a complex number is the angle between the positive real axis and the line joining the origin and the complex number.
– It is represented as a point in the complex plane.
– The argument can be defined geometrically as the polar angle from the positive real axis to the vector representing the complex number.
– Algebraically, it can be defined as any real quantity that satisfies the equation z = r(cos φ + i sin φ), where r is the modulus of the complex number.
– The argument of a non-zero complex number has multiple possible values due to the periodicity of sin and cos.
– The complex argument can be defined algebraically in terms of complex roots.
– It can be expressed as the limit of n times the imaginary part of the nth root of z divided by the modulus of z.
– This definition eliminates the need for difficult-to-compute functions and inherits the principal branch of the square root.
Principal Value and Notation
– The principal value of the argument is the well-defined value within the interval (-π, π].
– It represents an angle of up to half a complete circle from the positive real axis in either direction.
– The principal value is commonly denoted as Arg.
– There are many choices for the argument due to the possibility of circling the origin any number of times.
– The principal value is chosen to ensure a unique and consistent representation.
– The principal value of the argument is sometimes denoted with a capitalized initial letter, such as Arg.
– The notation for the argument may vary in different texts, with arg and Arg being used interchangeably.
– The set of all possible values of the argument can be written in terms of the principal value as arg(z) = n ∈ Z.
Computing from the Real and Imaginary Part
– The principal value of the argument can be computed from the real and imaginary parts of a complex number.
– The two-argument arctangent function, atan2, is commonly used for this computation.
– The atan2 function returns a value in the range (-π, π].
– Alternative definitions of the argument using arctan are valid only when certain conditions are met.
– The atan2 function is available in many programming languages.
Relationship between Argument and Trigonometry
– The argument of a complex number can be related to trigonometric functions.
– The real part of a complex number can be expressed as the magnitude multiplied by the cosine of its argument.
– The imaginary part of a complex number can be expressed as the magnitude multiplied by the sine of its argument.
– The argument of a complex number can be used to find its magnitude and phase.
– Trigonometric identities can be applied to manipulate and simplify expressions involving arguments.
Applications of Argument in Complex Analysis and Signal Processing
– The argument plays a crucial role in understanding the behavior of complex functions.
– It helps in determining the location of zeros, poles, and singularities of complex functions.
– The argument principle is a theorem in complex analysis used to count the number of zeros and poles inside a closed curve.
– The argument is used to define the logarithmic function for complex numbers.
– The argument is utilized in the study of contour integration and the evaluation of complex integrals.
– In signal processing, the argument of a complex number can represent the phase of a signal.
– The phase information is essential for analyzing and manipulating signals.
– The argument is used in Fourier analysis to decompose a signal into its frequency components.
– The phase spectrum of a signal can be obtained by calculating the argument of its Fourier transform.
– The argument is utilized in applications such as audio processing, image processing, and telecommunications. Source: https://en.wikipedia.org/wiki/Argument_(complex_analysis)
In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign.
When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval (−π, π]. In this article the multi-valued function will be denoted arg(z) and its principal value will be denoted Arg(z), but in some sources the capitalization of these symbols is exchanged.
