series whose coefficients are the generalized binomial coefficients (If , this product is an empty product and has value 1.) It converges for for any real or complex number . When , this is essentially the infinite geometric series mentioned in the previous section. The special cases and give the square root function and its inverse: When only the linear term is retained, this simplifies to the binomial approximation. Trigonometric functions The usual trigonometric functions and their inverses have the following Maclaurin series: All angles are expressed in radians. The numbers appearing in the expansions of are the Bernoulli numbers. The in the expansion of are Euler numbers. Hyperbolic functions The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions: The numbers appearing in the series for are the Bernoulli numbers. Polylogarithmic functions The polylogarithms have these defining identities: The Legendre chi functions are defined as follows: And the formulas presented below are called inverse tangent integrals: In statistical thermodynamics these formulas are of great importance. Elliptic functions The complete elliptic integrals of first kind K and of second kind E can be defined as follows: The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series: The regular partition number sequence P(n) has this generating function: The strict partition number sequence Q(n) has that generating function: Calculation of Taylor series Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. Particularly convenient is the use of computer algebra systems to calculate Taylor series. First example In order to compute the 7th degree Maclaurin polynomial for the function , one may first rewrite the function as . The Taylor series for the natural logarithm is (using the big O notation) and for the cosine function . The latter series expansion has a zero constant term, which enables us to substitute the second series into the