In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other. Characterizations The six most common definitions of the exponential function for real are: Define by the limit Define as the value of the infinite series (Here denotes the factorial of . One proof that is irrational uses a special case of this formula.) Define to be the unique number such that This is as the inverse of the natural logarithm function, which is defined by this integral. Define to be the unique solution to the initial value problem (Here, denotes the derivative of .) The exponential function is the unique function with and for all and that satisfies any one of the following additional conditions: For the uniqueness, one must impose some additional condition like those above, since otherwise other functions can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg. One could also replace and the "additional condition" with the single condition . Let be the unique positive real number satisfying This limit can be shown to exist. Then define to be the exponential function with this base. This definition is particularly suited to computing the derivative of the exponential function. Larger domains One way of defining the exponential function for domains larger than the domain of real numbers is to first define it for the domain of real numbers using one of the above characterizations and then extend it to larger domains in a way which would work for any analytic function. It is also possible to use the characterisations directly for the larger domain, though some problems may arise. (1), (2), and (4) all make sense for arbitrary Banach algebras. (3) presents a problem for complex numbers, because there are non-equivalent paths along which one could integrate, and (5) is not sufficient. For example, the function f defined (for x and y real) as satisfies the conditions in (5) without being the exponential function of . To make (5) sufficient for the domain of complex numbers, one may either stipulate that there