Function decomposition in a row is called its representation in the form of a limit of the infinite sum: F(z) = ∑fn(z) where n = 1 … ∞, and the fn (z) functions are called members of a functional row.
Instruction
1. For a number of reasons power series, that is ranks which formula has an appearance most of all are suitable for decomposition of functions: f(z) = c0 + c1 (z - a) + c2 (z - a) ^2 + c3 (z - a) ^3 + … + cn (z - a) ^n + … Number a is called in this case the center of a row. In particular, it can be equal to zero.
2. The power series have convergence radius. Radius of convergence is such number R that if |z-a| R it disperses, at |z-a| = R are possible both cases. In particular, the radius of convergence can be equal to infinity. In this case a row agrees on all valid axis.
3. It is known that the power series can be differentiated term by term, and the sum of the received row is equal derivative of the sum of an initial row and has the same radius of convergence. Based on this theorem, the formula called Taylor's number was removed. If the f(z) function can be spread out in power series with center a, then this row will have an appearance: f(z) = f(a) + f ′ (a) * (z - a) + (f ′′ (a)/2!) * (z - a) ^2 + … + (fn(a)/n!) * (z - a) ^n, where fn(a) — value of derivative n-go of an order from f(z) in a point. Designation n! (it is read "en a factorial") replaces the work of all integers from 1 to n.
4. If a = 0, then Taylor's number turns into the private option called Makloren's number: f(z) = f(0) + f ′ (0) *z + (f ′′ (0)/2!) *z^2 + … + (fn(0)/n!) *z^n.
5. For example, let Maklorena is required to spread out in a row the e^x function. As (e^x) ′ = e^x, all coefficients of fn (0) will be equal to e^0 = 1. Therefore, the general coefficient of the necessary row equals 1/n!, and the formula of a row looks as follows: e^x = 1 + x + (x^2)/2! + (x^3)/3! + … + (x^n)/n! + … Radius of convergence of this row is equal to infinity, that is he meets at any value x. In particular, for x = 1 this formula turns into the known expression for calculation of e.
6. Calculation for this formula can be easily executed even manually. If n-oye composed is already known, then to find (n + 1) - oye, it is enough to increase of it on x and to divide on (n + 1).