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Arg (mathematics)
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In mathematics, arg is a function operating on complex numbers (visualised as a flat plane), and intuitively gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point (that is, the angle between the half-lines of the position vector representing the number and the positive real axis).
The names amplitude or phase are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any (non-zero) complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2p radians (a complete circle) are the same.
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In mathematics, arg is a function operating on complex numbers (visualised as a flat plane), and intuitively gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point (that is, the angle between the half-lines of the position vector representing the number and the positive real axis).
Definition
Arguments are defined in two equivalent ways:
- Geometrically, in relation to an Argand diagram, arg z is the angle f, between the positive real axis and the vector representing z.
- Algebraically, an argument of the complex number z = x + iy is any real quantity f where:for some positive real . (The quantity is the modulus of , written .)
The names amplitude or phase are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any (non-zero) complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2p radians (a complete circle) are the same. Similarly, from the periodicity of and , the second definition also has this property.
Principal value
Because a complete rotation around 0 leaves a complex number unchanged, there are many choices which could be made for by circling the origin any number of times. This is shown in figure 2, a representation of the multi-valued (set-valued) function, where a vertical line cuts the surface at heights representing all the possible choices of angle for that point.
When a well-defined function is required then the usual choice, known as the principal value, is the value in the open-closed interval (-p, p], that is from -p to p radians, excluding -p itself (−180 to +180 degrees). This represents an angle of up to half a complete circle from the positive real axis in either direction, the angle is constrained to lie between -p and p radians. This portion of the surface is shown hatched in red in figure 2, and projected onto the plane in figure 3.
The principal value sometimes has the initial letter capitalized as in Arg z, especially when a general version of the argument is also being considered. Note that notation varies, so arg and Arg may be interchanged in different texts.
Some authors define the range of the principal value as being in the closed-open interval [0, 2p).
The set of all possible values of the argument can be written in terms of Arg as:
.
Covering space
In informal situations, arg may be left not well-defined, for instance arg z(t) where z depends on a parameter t may change by 2p every time z goes around the origin. This idea can be made more precise by considering z(t) as being defined not on the complex plane but on a covering space. Polar coordinates excluding the origin and with an unconstrained angle provide such a space, in this case arg is defined by:
The covering space has as base space the punctured complex plane. This is equivalent to the product of a positive non-zero radius and an angle on a unit circle that is:
The principal value Arg then maps the unit circle component of this representation to the interval (-p,p].
Computation
The principal value Arg of a complex number given as x+iy can be calculated using the tangent half-angle formula, it is defined over the complex plane but excluding the origin:
This version of Arg is not stable enough for numerical use but can be used in symbolic calculation. In many programming libraries there is a function called atan2 which performs an equivalent computation.
Many texts say the value is given by arctan(y/x) and this is correct when x > 0, however this has problems requiring a large number of special cases when x is not positive.
For the variant where Arg is defined to lie in the interval [0, 2p), the value can be got by adding 2p to the value above when it is negative.
Identities
One of the main motivations for defining the principal value arg is to be able to write complex numbers in modulus-argument form (the modulus of z = x + iy is |z| = √(x2 + y2), the length of the vector on the Argand diagram). Hence for any complex number z,
.
This is only really valid if z is non-zero but can be considered as valid also for z = 0 if arg(0) is considered as being an indeterminate form rather than as being undefined.
Some further identities follow. If z1 and z2 are two non-zero complex numbers then
and
If z ? 0 and n is any integer then
Example
Bibliography
External links
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