Sinh

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

sinh(((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

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

Symmetries and periodicities

Parity

sinh(z) is an odd function.

Mirror symmetry

Periodicity

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

Poles and essential singularities

The function sinh(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

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

Branch cuts

The function sinh(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

Exponential Fourier series

Asymptotic series expansions

Other series representations

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Product representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

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

Powers involving the direct function

Sums of powers involving the direct function

Related transformations

Identities

Functional identities

Univariate functional identities

Biivariate functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Differentiation

Low-order differentiation

Symbolic differentiation

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 algebraic functions

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, exponential and algebraic 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

Definite integration

For the direct function itself

Involving the direct function

Involving related functions

Integral transforms

Laplace transforms

Summation

Finite summation

Infinite summation

Products

Finite products

Infinite products

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric0F1

Involving HypergeometricPFQ

Brychkov Yu.A. (2005)

Through Meijer G

Classical cases for the direct function itself

Classical cases for powers of Cosh

Classical cases involving Exp

Classical cases involving Sin

Classical cases involving Cos

Classical cases involving ArcSinh in the arguments

Classical cases involving ArcCsch in the arguments

Classical cases involving ArcCosh in the arguments and unit step UnitStep

Classical cases involving ArcSech in the arguments and unit step UnitStep

Classical cases involving ArcTanh in the arguments and unit step UnitStep

Classical cases involving ArcTanh in the arguments and Abs

Classical cases involving ArcCoth in the arguments and Abs

Classical cases involving ArcTanh in the arguments and Sign

Classical cases involving ArcCoth in the arguments and Sign

Classical cases involving Bessel BesselI

Classical cases involving Cosh, BesselI

Classical cases involving Bessel BesselK

Classical cases involving ParabolicCylinderD

Classical cases involving Exp and Hermite HermiteH

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Classical cases involving  Exp and Hypergeometric1F1

Classical cases involving  Exp and Hypergeometric1F1Regularized

Classical cases involving  Exp and hypergeometric HypergeometricU

Classical cases involving  Exp and Laguerre LaguerreL

Classical cases involving WhittakerM WhittakerM

Classical cases involving WhittakerW WhittakerW

Generalized cases for the direct function itself

Generalized cases for powers of Sinh

Generalized cases involving Cos

Generalized cases involving Sin

Generalized cases involving ArcSinh in the arguments

Generalized cases involving ArcCsch in the arguments

Generalized cases involving ArcCosh in the arguments and unit step UnitStep

Generalized cases involving ArcSech in the arguments and unit step UnitStep

Generalized cases involving ArcTanh in the arguments and unit step UnitStep

Generalized cases involving ArcTanh in the arguments and Abs

Generalized cases involving ArcCoth in the arguments and Abs

Generalized cases involving ArcTanh in the arguments and Sign

Generalized cases involving ArcCoth in the arguments and Sign

Generalized cases involving AiryAi

Generalized cases involving AiryAiPrime

Generalized cases involving AiryBi

Generalized cases involving AiryBiPrime

Generalized cases involving Bessel BesselI

Generalized cases involving Cosh, BesselI

Generalized cases involving Bessel BesselK

Generalized cases involving ParabolicCylinderD

Generalized cases involving Exp and Hermite HermiteH

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Through other functions

Involving Bessel functions

Involving Jacobi functions

Involving Mathieu functions

Involving some elliptic-type 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 Cosh

Involving Tanh

Involving Coth

Involving Csch

Involving Sech

Inequalities

Zeros

Theorems

Bogoliubov transformation

Bogoliubov transformation of thermo field theory a⟶cosh(α) a -sinh(α) (SuperPlus[(OverTilde[a])]) , (OverTilde[a])⟶cosh(α) (OverTilde[a]) -sinh(α) (SuperPlus[a]) , where a is the creation operator and (OverTilde[a]) is its tilde version, leaves the free Hamiltonian invariant.

History

V. Riccati (1757); D. Foncenex (1759); J.H. Lambert (1768).

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

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