2cosh2z

2cosh2z is a mathematical expression involving the hyperbolic cosine function. The hyperbolic cosine of a complex number z, denoted as cosh(z), is defined as (e^z + e^-z) / 2. Therefore, 2cosh2z is equivalent to 2 * (e^(2z) + e^(-2z)) / 2. This simplifies to e^(2z) + e^(-2z).

The expression 2cosh2z can also be related to other hyperbolic functions through identities. For instance, it is equal to 2 * (2cosh^2(z) - 1) using the double-angle formula for hyperbolic cosine, which is cosh(2z) = 2cosh^2(z) - 1. Alternatively, it can be expressed as 2 * (1 + 2sinh^2(z)), derived from the identity cosh(2z) = 1 + 2sinh^2(z).

In the realm of complex analysis, 2cosh2z plays a role in various series expansions and integral evaluations. Its properties, such as periodicity and its behavior under differentiation and integration, are consistent with those of hyperbolic functions. The derivative of 2cosh2z with respect to z is 4sinh2z, and its integral is sinh(2z) + C, where C is the constant of integration. This expression is fundamental in understanding the behavior of solutions to certain differential equations and in Fourier analysis involving hyperbolic components.