Understanding the Hyperbolic Secant Function | Definition, Properties, and Applications

sech x

In mathematics, the term “sech x” refers to the hyperbolic secant function

In mathematics, the term “sech x” refers to the hyperbolic secant function. The hyperbolic functions are analogs of the trigonometric functions but are related to exponential functions.

The hyperbolic secant function, sech x, is defined as the reciprocal of the hyperbolic cosine function, cosh x. Mathematically, sech x can be expressed as:

sech x = 1 / cosh x

The hyperbolic cosine function, cosh x, is defined as:

cosh x = (e^x + e^(-x)) / 2

where e represents Euler’s number approximately equal to 2.71828.

By substituting the expression for cosh x into the definition of sech x, we can expand it further:

sech x = 1 / ((e^x + e^(-x)) / 2)
= 2 / (e^x + e^(-x))

The hyperbolic secant function, sech x, is symmetric about the y-axis, meaning sech(-x) = sech x. It has a range between 1 (when x approaches negative or positive infinity) and 0 (at x = 0).

The sech function commonly arises in various areas of mathematics, including calculus, differential equations, and physics (particularly in the study of wave functions and quantum mechanics). It has properties similar to the trigonometric secant function, but is defined in terms of the hyperbolic cosine function instead of the cosine function.

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