Search results
Results from the WOW.Com Content Network
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β. Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas ...
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 ...
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number , except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°).
A summation method is any method for assigning sums to divergent series in a way that systematically extends the classical notion of the sum of a series. Summation methods include Cesàro summation, generalized Cesàro (,) summation, Abel summation, and Borel summation, in order of applicability to increasingly divergent series.
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function on the interval [,], which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula. Sum