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Arcsinh is the inverse function of the hyperbolic sine function. It is denoted by arcsinh or sinh^-1 and is defined as the inverse of sinh(x), which is the ratio of the hyperbolic sine of x to x. The hyperbolic sine function is defined as sinh(x) = (ex - e^(-x)) / 2.

The arcsinh function is defined as arcsinh(y) = ln(y + sqrt(y^2 + 1)), where ln is the natural logarithm. This definition is valid for any real y and is consistent with the original definition of sinh(x).

As an inverse function, arcsinh(y) gives the value of x such that sinh(x) = y. The range of the function is all real numbers, and the domain is y >= 0. When y is negative, the value of arcsinh(y) is undefined in the real number system. However, it can be extended to the complex number system by considering the complex logarithm.

The arcsinh function has multiple applications in mathematics and physics, particularly in the field of calculus and differential equations. It is used to solve equations involving the hyperbolic sine and cosine functions, and it is also related to integrals and derivatives of these functions.

A key property of the arcsinh function is its relationship with the standard inverse trigonometric functions. Specifically, arcsinh(x) and asinh(x) are equivalent, where asinh(x) is the inverse function of sinh(x) on the unit hyperbola. This relationship can be useful in algebraic manipulations and simplifications involving these functions.