Tanh

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

tanh(((n i π)/m)) can be expressed using only square roots if n∈ℤ and m is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

Values at infinities

General characteristics

Domain and analyticity

tanh(z) is an analytical function of z which is defined over the whole complex z-plane with the exception of countably many points z=π i/2+k π i/;k∈ℤ.

Symmetries and periodicities

Parity

tanh(z) is an odd function.

Mirror symmetry

Periodicity

tanh(z) is a periodic function with period π i.

Poles and essential singularities

The function tanh(z) has an infinite set of singular points:
a) z==π i/2+π k i/;k∈ℤ are the simple poles with residues 1;
b) z== (OverTilde[∞]) is an essential singular point.

Branch points

The function tanh(z) does not have branch points.

Branch cuts

The function tanh(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==((π i)/2)

q-series

Dirichlet series

Asymptotic series expansions

Other series representations

Integral representations

On the real axis

Of the direct function

Limit representations

Continued fraction representations

Differential equations

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Argument involving inverse trigonometric and hyperbolic functions

Addition formulas

Half-angle formulas

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Roger Germundsson

Products, sums, and powers of the direct function

Products of the direct function

Products involving the direct function

Sums of the direct function

Sums involving the direct function

Powers of the direct function

Sums of powers involving the direct function

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Symbolic differentiation

Victor Adamchik (2005)

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and a power function

Involving functions of the direct function and exponential function

Involving functions of the direct function, exponential and a power functions

Involving functions of the direct function and trigonometric functions

Involving functions of the direct function, trigonometric and a power functions

Involving functions of the direct function, trigonometric and exponential functions

Involving functions of the direct function, trigonometric, exponential and a power functions

Involving functions of the direct function and hyperbolic functions

Involving functions of the direct function, hyperbolic and a power functions

Involving functions of the direct function, hyperbolic and exponential functions

Involving functions of the direct function, hyperbolic and trigonometric functions

Involving functions of the direct function, hyperbolic, exponential and a power functions

Involving functions of the direct function, hyperbolic, exponential and trigonometric functions

Involving functions of the direct function, hyperbolic, trigonometric and a power functions

Involving functions of the direct function, hyperbolic, exponential, trigonometric and a power functions

Definite integration

For the direct function itself

Involving related functions

Summation

Finite summation

Infinite summation

Representations through more general functions

Through hypergeometric functions

Brychkov Yu.A. (2005)

Through other functions

Involving Jacobi functions

Involving Mathieu functions

Representations through equivalent functions

With inverse function

With related functions

Involving Exp

Involving Sin

Involving Cos

Involving Tan

Involving Cot

Involving Csc

Involving Sec

Involving Sinh

Involving Cosh

Involving Coth

Involving Csch

Involving Sech

Involving trigonometric and hyperbolic functions

Inequalities

Theorems

Solution of the associated Camassa-Holm equation

The function p(x,t)==(h+(λ/h)) (tanh^2)((Sqrt[(1/λ)+(1/(h^2))]) (x+h t λ))-(λ/h) is a solution of the associated Camassa–Holm equation

((∂p(x,t))/(∂t))==(((p(x,t)))^2) (∂/(∂x))((1/4) p(x,t) (((∂^2)log(p(x,t)))/(∂x ∂t))-(1/2) (((p(x,t)))^2)).

History

– L'Abbe Sauri (1774)

Copyright