ArcTan2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed x

For fixed y

General characteristics

Domain and analyticity

(tan^(-1))(x,y) is an analytical function of x and y, which is defined over (ℂ^2).

Symmetries and periodicities

Parity

(tan^(-1))(x,y) is an even function.

Mirror symmetry

Periodicity

No periodicity

Homogeneity

Poles and essential singularities

With respect to x

The function (tan^(-1))(x,y) does not have poles and essential singularities with respect to x.

With respect to y

The function (tan^(-1))(x,y) does not have poles and essential singularities with respect to y.

Branch points

With respect to y

With respect to x

Branch cuts

With respect to y

For fixed x, the branch cuts lie along three curves: (y==(Sqrt[t-(x^2)]), -∞<t≤ 0), (y==-(Sqrt[t-(x^2)]), -∞<t≤ 0), and (y==θ(-Re(x)) i x t, -1≤t≤1).

With respect to x

For fixed y, the branch cuts lie along three curves: (x==(Sqrt[t-(y^2)]), -∞<t≤ 0), (x==-(Sqrt[t-(y^2)]), -∞<t≤ 0), and (x==sign(Im(y)) i y t, 1≤t<∞).

Analytic continuations

The analytic continuation of (tan^(-1)) has infinitely many sheets; the values of ((OverTilde[tan])^(-1)) are ((OverTilde[tan])^(-1))(z)=(tan^(-1))(z)+k π /;k∈ℤ.

Series representations

Generalized power series

Expansions at x==0

Expansions at x==∞

Expansions at y==0

Expansions at y==∞

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Partial differential equations

Transformations

Transformations and argument simplifications

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

With respect to x

With respect to y

Symbolic differentiation

With respect to x

With respect to y

Fractional integro-differentiation

With respect to x

With respect to y

Integration

Indefinite integration

With respect to x

With respect to y

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through other functions

Representations through equivalent functions

With related functions

Involving Log

Involving ArcTan

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