Introduction to the Hyperbolic Sine Function
Defining the hyperbolic sine function
The hyperbolic sine function is an old mathematical function. It was first used in the works of V. Riccati (1757), D. Foncenex (1759), and J. H. Lambert (1768).
The hyperbolic sine function is easily defined as the half difference of two exponential functions in the points and :
After comparison with the famous Euler formula for sine (), it is easy to derive the following representation of the hyperbolic sine through the circular sine:
This formula allows the derivation of all the properties and formulas for the hyperbolic sine from the corresponding properties and formulas for the circular sine.
The following formula can sometimes be used as an equivalent definition of the hyperbolic sine function:
This series converges for all finite numbers .
A quick look at the hyperbolic sine function Here is a graphic of the hyperbolic sine function for real values of its argument .
Representation through more general functions
The function is a particular case of more complicated mathematical functions. For example, it is a special case of the generalized hypergeometric function with the parameter at , multiplied by :
It is also a particular case of the modified Bessel function with the parameter , multiplied by :
Other Bessel functions can also be expressed through the hyperbolic sine function for similar values of the parameter:
Struve functions can also degenerate into the hyperbolic sine function for a similar parameter value:
But the function is also a degenerate case of the doubly periodic Jacobi elliptic functions when their second parameter is equal to or :
Finally, the function is the particular case of another class of functions—the Mathieu functions:
Definition of the hyperbolic sine function for a complex argument In the complex ‐plane, the function is defined by the same formula used for real values: Here are two graphics showing the real and imaginary parts of the hyperbolic sine function over the complex plane.
The bestknown properties and formulas for the hyperbolic sine function
Values in points
The values of the hyperbolic sine function for special values of its argument can be easily derived from the corresponding values of the circular sine in special points of the circle:
The values at infinity can be expressed by the following formulas:
General characteristics
For real values of argument , the values of are real.
In the points , the values of are algebraic. In several cases, they can even be rational numbers, 0, or 1, multiplied by . Here are some examples:
The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.
The function is an entire analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an essential singular point at . It is a periodic function with the period :
The function is an odd function with mirror symmetry:
Differentiation
The derivatives of have simple representations using either the function or the function:
Ordinary differential equation
The function satisfies the simplest possible linear differential equation with constant coefficients:
The complete solution of this equation can be represented as a linear combination of and with arbitrary constant coefficients and :
The function also satisfies the first‐order nonlinear differential equation:
Series representation
The function has a simple series expansion at the origin that converges in the whole complex ‐plane:
Product representation
The following infinite product representation for clearly illustrates that at :
Indefinite integration
Indefinite integrals of expressions involving the hyperbolic sine function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:
The last integral cannot be evaluated in closed form using the known classical special functions for arbitrary values of parameters and .
Definite integration
Definite integrals that contain the hyperbolic sine are sometimes simple as shown in the following example:
Some special functions can be used to evaluate more complicated definite integrals. For example, gamma and polygamma functions, are needed to express the following integrals:
Integral transforms
Numerous formulas for integral transforms from circular sine functions cannot be easily converted into corresponding formulas with the hyperbolic sine function because the hyperbolic sine grows exponentially at infinity. This holds for the Fourier cosine and sine transforms, and for Mellin, Hilbert, Hankel, and other transforms.
The exceptional case is the Laplace transform that itself includes the exponential function in the kernel:
Finite summation
The following finite sum of the hyperbolic sine can be expressed using the hyperbolic sine function:
Infinite summation
The following infinite sum can be expressed using elementary functions:
Finite products
The following finite products of the hyperbolic sine can be expressed using elementary functions and formulas:
Infinite products
The following infinite product can be expressed using the hyperbolic sine function:
Addition formulas
The hyperbolic sine of a sum can be represented by the rule: "the hyperbolic sine of a sum is equal to the product of the hyperbolic sine by the hyperbolic cosine plus the hyperbolic cosine by the hyperbolic sine." A similar rule is valid for the hyperbolic sine of the difference:
Multiple arguments
In the case of multiple arguments , , …, the function can be represented as the finite sum of terms that include powers of the hyperbolic sine and cosine:
The function can also be represented as the finite product including the hyperbolic sine of the linear argument of :
Halfangle formulas
The hyperbolic sine of the half‐angle can be represented by the following simple formula that is valid in half of the horizontal strip:
To make this formula correct for all complex , a complicated prefactor is needed:
where contains the unit step, real part, imaginary part, and the floor functions.
Sums of two direct functions
The sum of two hyperbolic sine functions can be described by the rule: "the sum of hyperbolic sines is equal to the doubled hyperbolic cosine of the half‐difference multiplied by the hyperbolic sine of the half‐sum". A similar rule is valid for the difference of two hyperbolic sines:
Products involving the direct function
The product of two hyperbolic sine functions or the product of the hyperbolic sine and hyperbolic cosine have the following representations:
Powers of the direct function
The integer powers of the hyperbolic sine functions can be expanded as finite sums of hyperbolic cosine (or sine) functions with multiple arguments. These sums include binomial coefficients:
These formulas can be combined into the following formula:
Inequalities
The most famous inequality for the hyperbolic sine function can be described by the following formula:
Relations with its inverse function
There is a simple relation between the function and its inverse function :
The second formula is valid at least in the horizontal strip . Outside this strip, a much more complicated relation (that contains the unit step, real part, imaginary part, and the floor functions) holds:
Representations through other hyperbolic functions
Hyperbolic sine and cosine functions are connected by a very simple formula that contains the linear function in the argument:
Another famous formula, connecting and , is expressed in the analog of the well‐known Pythagorean theorem:
The last restriction on can be removed, but the formula will get a complicated coefficient that contains the unit step, real part, imaginary part, and the floor function and c(z) ⩵ 1:
The hyperbolic sine function can also be represented using other hyperbolic functions by the following formulas:
Representations through hyperbolic functions
The hyperbolic sine function has representations using the other hyperbolic functions:
Applications
The hyperbolic sine function is used throughout mathematics, the exact sciences, and engineering.
