A Detailed Note on the Argument of z

This angle is  written as φ and is referred to as the angle between the positive real axis and the line that joins the origin and the complex number z. 

It is a multi-valued function that operates on the complex integers that are not zero. 

For the purpose of defining a function with a single value, the primary value of the argument (sometimes referred to as Arg z) is utilised. 

It is frequently selected because it is the only possible value for the argument that can be found inside the interval [–π,π].

Definition

The following are two synonyms for the definition of the argument of the complex number z = x + iy, which is denoted by the symbol arg(z):

Geometrically speaking, in the complex plane, it is represented as the 2D polar angle   φ between the positive real axis and the vector that stands for z.

 If the angle is measured in a counterclockwise direction, the resulting numeric value will be positive.

 The angle is measured in radians.

In terms of algebra, it can be represented as any real quantity   φ such that  z=r(cos φ +isin φ)=reiφ for some positive real r. 

The modulus (or absolute value) of z is the quantity r, and it is symbolised by the symbol |z|. 

The formula for r is as follows: 

 r=√x2 +y2

There is a degree of semantic overlap between the terms magnitude, which refers to the modulus, and phase, which refers to the argument.

According to both definitions, it is clear that the argument of any non-zero complex number can take on a wide variety of different values. 

To begin, when viewed as a geometrical angle, it is obvious that rotations of a whole circle do not affect the point; 

Consequently, angles that differ by an integer multiple of 2π radians (a complete circle) are the same.

In a similar manner, this property is present in the second formulation as a result of the periodicity of sin and cos. 

In most cases, the meaning of the argument of zero is not specified.

Principal value

Because a full revolution around the origin does not alter a complex number, there are many different options available for   φ. 

These options can be accessed by circling the origin any number of times. 

 Function  f(x,y)=arg(x+iy), in which a vertical line cuts the surface at heights representing all of the possible choices of angle for that point.

This value is referred to as the principal value, and it ranges from to radians, excluding radians themselves (or, equivalently, from -180 to +180 degrees, excluding 180° itself). 

This indicates an angle of up to half a complete circle, in either direction, measured counterclockwise from the positive real axis.

Some writers place the range of the principal value in the closed-open interval [0, 2π], and this is how they define that range.

Notation

Sometimes the initial letter of the major value is capitalised, such as in the case of Arg z, and this is especially the case when a broader version of the argument is also being examined.

 It is important to keep in mind that notation varies, therefore the terms arg and Arg may be substituted for one another in different books.

When expressed in terms of Arg, the complete set of all possible values for the argument can be stated as:

 Arg(z) =arg(z)+2πn

 Where, n € Z.

Calculating based on both the real and the imaginary component

If a complex number is known in terms of the real and imaginary parts that make up the number, then the function that calculates the principal value Arg is referred to as the arctangent function with two arguments, or atan2:

Arg (x+iy) = atan2 (y,x) 

The atan2 function, which also goes by the name arctan2 and has a few more synonyms, can be found in the math libraries of a wide variety of programming languages. 

It returns a value that is typically in the range [-π,π].

As y/x represents the slope, and arctan is used to convert slope to angle, several texts assert that the value may be calculated using arctan(y/x). 

This is true only when x is more than zero, at which point the quotient is defined and the angle falls somewhere between – π/2 and π/2.

Nevertheless, extending this definition to circumstances in which x is not positive is a relatively difficult process. 

To be more specific, one can define the principal value of the argument independently on the two half-planes x > 0 and x 0 (split into two quadrants if one needs a branch cut on the negative x-axis), y > 0, and y 0, and then patch together the results of those definitions.

Conclusion

In order to be able to write complex numbers in modulus-argument form, it is necessary to define the principle value Arg as one of the primary motives for doing so. 

Therefore, in the case of any complex number z.

z= |z| ei Arg z

This is only truly valid if z is a non-zero value; nevertheless, it can be regarded as valid for z = 0 if Arg(0) is considered to be an indeterminate form rather than a form that is undefined. 

Z must be non-zero for this to be valid.