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Hyperbolic functions are analogues of trigonometric functions, but defined using the hyperbola. Tanh is the hyperbolic tangent, which is the ratio of the hyperbolic sine and cosine functions. Learn how to define, derive, and use tanh in various contexts.
The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its probability density function is proportional to its characteristic function.
Learn about the inverse hyperbolic functions, which are analogous to the inverse circular functions and have applications in hyperbolic geometry and differential equations. Find definitions, formulas, graphs, and principal values of the inverse hyperbolic cosine (arccosh) and other functions.
Learn how to evaluate the integral of the secant function using various methods and expressions, such as trigonometric, hyperbolic, and complex forms. Find the history, proofs, and applications of this integral, especially for the Mercator projection.
FWHM is the difference between the two values of the independent variable at which the dependent variable is half of its maximum value. It is applied to various phenomena such as pulse waveforms, spectral width, signal processing and antenna beam width.
Learn how to derive and apply the integral of secant cubed, a challenging and frequent indefinite integral of elementary calculus. See different methods, such as integration by parts, partial fractions, and hyperbolic functions.
The probability density function is the partial derivative of the cumulative distribution function: (;,) = (;,) = / (+ /) = (() / + / ()) = ().When the location parameter μ is 0 and the scale parameter s is 1, then the probability density function of the logistic distribution is given by
The Hubbert curve is a logistic distribution curve that approximates the production rate of a resource over time. It is used to predict the depletion of various natural resources, such as oil, gas, coal, and water, and has been criticized for its assumptions and limitations.