Sech

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

sech(((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

sech(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Parity

sech(z) is an even function.

Mirror symmetry

Periodicity

sech(z) is a periodic function with period 2π i.

Poles and essential singularities

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

Branch points

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

Branch cuts

The function sech(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

Product representations

Limit 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

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 the direct function

Summation

Finite summation

Infinite summation

Products

Finite products

Infinite products

Representations through more general functions

Through hypergeometric functions

Brychkov Yu.A. (2005)

Through Meijer G

Classical cases for the direct function itself

Generalized cases for the direct function itself

Through other functions

Involving Bessel functions

Involving Jacobi functions

Involving Mathieu functions

Involving some hypergeometric-type 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 Tanh

Involving Coth

Involving Csch

Involving trigonometric and hyperbolic functions

Inequalities

Theorems

Localized soliton of the Korteweg-de Vries equation

The function u(x,t)==(sech^2)(((x-(t/3))/(2 (Sqrt[3])))) is a localized soliton of the Korteweg-de Vries equation

((∂u(x,t))/(∂t))+((∂u(x,t))/(∂x)) u(x,t)+(((∂^3)u(x,t))/(∂((x )^3)))==0.

History

The function sech is encountered often in mathematics and the natural sciences.

Copyright