Curvarum. It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor, after whom the series are now named. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid-18th century. Analytic functions If is given by a convergent power series in an open disk centred at in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for in this region, is given by a convergent power series Differentiating by the above formula times, then setting gives: and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at if and only if its Taylor series converges to the value of the function at each point of the disk. If is equal to the sum of its Taylor series for all in the complex plane, it is called entire. The polynomials, exponential function , and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if is far from . That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions include: The partial sums (the Taylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included. Differentiation and integration of power series can be performed term by term and is hence particularly easy. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available. The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). Algebraic operations can be done readily on the power series