cosh2z

The hyperbolic cosine squared function, often denoted as cosh^2(z) or cosh2z, is a mathematical function derived from the hyperbolic cosine function, cosh(z). The hyperbolic cosine is defined as cosh(z) = (e^z + e^-z) / 2. Therefore, cosh^2(z) is the result of squaring this expression: cosh^2(z) = ((e^z + e^-z) / 2)^2. Expanding this gives cosh^2(z) = (e^(2z) + 2 + e^(-2z)) / 4.

This function has several important properties and identities. It is closely related to the hyperbolic sine squared function, sinh^2(z), through the fundamental hyperbolic identity cosh^2(z) - sinh^2(z) = 1. From this, we can express cosh^2(z) in terms of sinh^2(z) as cosh^2(z) = 1 + sinh^2(z). Additionally, there is a double-angle identity for cosh(2z) which is cosh(2z) = 2cosh^2(z) - 1. Rearranging this gives an alternative expression for cosh^2(z): cosh^2(z) = (cosh(2z) + 1) / 2. This form is particularly useful in calculus for simplifying integrals.

The function cosh^2(z) is an entire function, meaning it is analytic over the entire complex plane. Its behavior is similar to that of cosine squared in trigonometry, exhibiting periodic behavior when considering the imaginary part of the input. Its real part, when z is real, grows exponentially as |z| increases.