first one and to easily omit terms of higher order than the 7th degree by using the big notation: Since the cosine is an even function, the coefficients for all the odd powers have to be zero. Second example Suppose we want the Taylor series at 0 of the function We have for the exponential function and, as in the first example, Assume the power series is Then multiplication with the denominator and substitution of the series of the cosine yields Collecting the terms up to fourth order yields The values of can be found by comparison of coefficients with the top expression for , yielding: Third example Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand as a Taylor series in , we use the known Taylor series of function : Thus, Taylor series as definitions Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm. In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution. Taylor series in several variables The Taylor series may also be generalized to functions of more than one variable with For example, for a function that depends on two variables, and , the Taylor series to second order about the point is where the subscripts denote the respective partial derivatives. A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as where is the gradient of evaluated at and is