ArcTan

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(tan^(-1))(z) is an analytical function of z, which is defined over the whole complex z-plane.

Symmetries and periodicities

Parity

(tan^(-1))(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function (tan^(-1))(z) does not have poles and essential singularities.

Branch points

Branch cut endpoints

At z=(OverTilde[∞]) two logarithmic branch points coincide in different directions: (tan^(-1))(z)∝((π (Sqrt[(z^2)]))/(2 z))+O((1/z))/;(|z|->∞). This results in z==(OverTilde[∞]) not being a branch point anymore; instead, two disconnected sheets arise.

Branch cuts

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 generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==0

Expansions at z==i

Expansions at z==-i

Expansions at z==∞

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Power of arguments

Products, sums, and powers of the direct function

Sums of the direct function

Differences of the direct function

Linear combinations of the direct function

Related transformations

Sums involving the direct function

Differences involving the direct function

Linear combinations involving the direct function

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Fractional integro-differentiation

Integration

Indefinite integration

For the direct function itself

Involving the direct function

Definite integration

For the direct function itself

Involving the direct function

Summation

Infinite summation

Integral transforms

Laplace transforms

Inverse Laplace transforms

Mellin transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Generalized cases for the direct function itself

Through other functions

Involving inverse Jacobi functions

Involving some elliptic-type functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

Involving ArcTan[Tan[z]]

Involving Tan[ArcTan[z]]

Involving ArcTan[Cot[z]]

With related functions

Involving Log

Involving ArcSin

Involving ArcCos

Involving ArcCot

Involving ArcCsc

Involving ArcSec

Involving ArcSinh

Involving ArcCosh

Involving ArcTanh

Involving ArcCoth

Involving ArcCsch

Involving ArcSech

Inequalities

Zeros

History

– J. Gregory (1671) found a series representation for arctan
- G. W. Leibniz (1674, 1682) rediscovered a series representation for arctan
– Joh. Bernoulli (1702) interpreted (tan^(-1)) as the corresponding Log expression
– L. Euler (1736); J. Herschel (1813)

The function (tan^(-1)) is encountered often in mathematics and the natural sciences.

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