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In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also ...
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar- , or with a superscript − 1 {\displaystyle {-1 ...
The (scaled and shifted) Gudermannian function is the cumulative distribution function of the hyperbolic secant distribution. A function based on the Gudermannian provides a good model for the shape of spiral galaxy arms.
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals . In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
A standard method of evaluating the secant integral presented in various references involves multiplying the numerator and denominator by sec θ + tan θ and then using the substitution u = sec θ + tan θ. This substitution can be obtained from the derivatives of secant and tangent added together, which have secant as a common factor. [6]
where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. [1] They are named after the French mathematician Louis Poinsot . Examples of the two types of Poinsot's spirals
where is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: = (). The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition.