Understanding the Coth(x) Function | Definition, Properties, and Applications

coth x

The function coth(x) is the hyperbolic cotangent function

The function coth(x) is the hyperbolic cotangent function. It is defined as the ratio of the hyperbolic cosine to the hyperbolic sine of a given angle x. Mathematically, coth(x) can be expressed as:

coth(x) = cosh(x) / sinh(x)

Here, cosh(x) represents the hyperbolic cosine function and sinh(x) represents the hyperbolic sine function.

To understand the behavior of coth(x), it is helpful to consider its graph. The coth(x) graph is symmetric about the y-axis, with vertical asymptotes at x = 0 and x = πi, where i represents the imaginary unit. This means that coth(x) tends towards positive or negative infinity as x approaches these asymptotes.

Additionally, the range of coth(x) excludes zero, which means that the function will never be equal to zero for any real value of x. As x increases or decreases, coth(x) approaches either positive or negative one asymptotically.

Some key properties of coth(x) include:

1. Even function: coth(-x) = coth(x)
2. Periodicity: coth(x + 2πi) = coth(x)
3. Relationship to sinh and cosh: coth(x) = 1 / tanh(x)

In practical applications, the coth(x) function is used in various fields such as physics, engineering, and finance to describe growth rates, heat transfer, and exponential decay, among others.

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