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CF. for. 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 ...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle . Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2. In the diagram, such a circle is tangent to the ...
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.
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [ 1][ 2] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial ...
Full width at half maximum. In a distribution, full width at half maximum ( FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y -axis which are half the maximum ...
Two-soliton solution to the KdV equation. In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an ...
A right triangle with sides relative to an angle at the point. Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that.
Integral. The integral of secant cubed is a frequent and challenging [1] indefinite integral of elementary calculus : where is the inverse Gudermannian function, the integral of the secant function . There are a number of reasons why this particular antiderivative is worthy of special attention: The technique used for reducing integrals of ...