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Power Series Solutions and Special Functions: Review of Power Series

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Power Series Solutions and Special Functions: Review of Power Series Power Series Solutions And Special Functions: Review of Power Series Pradeep Boggarapu Department of Mathematics BITS PILANI K K Birla Goa Campus, Goa September 16, 2015 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 1 / 48 Finding the general solution of a linear differential equation y 00 + P(x)y 0 + Q(x)y = R(x) depends on determining any two linear independent solutions of the homogeneous equation y 00 + P(x)y 0 + Q(x)y = 0 So far, we have a systematic procedure for constructing fundamental solutions (linear independent solutions of associated homogeneous equation), if the equation has constant coefficients (i.e., P(x) and Q(x) are constant functions). For the equations with variable coefficients we don't have any method to find fundamental solutions expect the case of knowing one solution where we use the known solution to find another solution via the formula y2(x) = v(x)y1(x) with Z 1 − R P(x)dx v(x) = 2 e dx y1 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 2 / 48 Examples: Let p; a; b; c and k are real constants, 2 00 0 2 2 1 Bessel's Differential Equation: x y + xy + (x − p )y = 0 2 00 0 2 Legender's Differential Equation: (1 − x )y − 2xy + p(p − 1)y = 0 00 0 3 Hermite Differential Equation: y − 2xy + 2py = 0 4 Guass Hypergeometric Differentila Equation: x(x − 1)y 00 + [c − (a + b + 1)x]y 0 − aby = 0 00 0 5 Laguerre's Differential Equations: xy + (1 − x)y + py = 0 00 2 6 Airy's Equation: y ± p xy = 0 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 3 / 48 In most of the cases the solutions of the above differential equations are beyond the elementary functions which are called as special functions. Many of the special functions find applications in connection with the partial differential equations of mathematical physics. Thare are also important in modern pure mathematics, through the theory of orthogonal expansions. For a larger class of linear diffrerential equations with variable coefficients such as above equations, we must search for solutions beyond the familiar elementary functions of calculus. The principal tool we need is the representation of a given function by a power series. Then, we assume that the solutions y have power series representations P1 n n=0 an(x − x0) , and then determine the coefficients an's so as to satisfy the differential equation similar to the method of undetermined coefficients. Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 4 / 48 Review of Power Series Power series about the point zero: It is an infinite series of the form 1 X n 2 3 anx = a0 + a1x + a2x + a3x + ··· (0.1) n=0 where a0; a1; a2;:::; an;::: are real constants. Power series about the point x0: It is an infinite series of the form 1 X n 2 3 an(x − x0) = a0 + a1(x − x0) + a2(x − x0) + a3(x − x0) + ··· (0.2) n=0 where a0; a1; a2;:::; an;::: are real constants. 1 X xn x x2 x3 Examples: = 1 + + + + ··· ; n! 1! 2! 3! n=0 1 X xn = 1 + x1 + x2 + x3 + ··· : n=0 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 5 / 48 Convergence of power series: The power series is said to converge at a Pn k point x if its n-th partial sum k=0 ak x converges that is to say that the n X k limit L = lim ak x exists. n!1 k=0 P1 n In this case the sum of the series is the limit i.e. L = n=0 anx and such points x are called points of convergence. Note that x = 0 is always a point of convergence of the power series (0.1). See the following examples 1 X xn x x2 x3 = 1 + + + + ··· ; (0.3) n! 1! 2! 3! n=0 1 X xn = 1 + x1 + x2 + x3 + ··· ; (0.4) n=0 1 X n!xn = 1 + x + 2!x2 + 3!x3 + ··· : (0.5) n=0 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 6 / 48 The first series converges for every value of x in R; second converges only for jxj < 1 and the third series diverges for all x 6= 0. The power series in x that behaves like third series are of no interest to us. Fact: The points of convergence of a power series (0.1) (or (0.2)) form an interval. Moreover there exists0 ≤ R ≤ 1 such that the power series (0.1) (or (0.2)) converges for all jxj < R (resp. jx − x0j < R) and diverges for all jxj > R (resp. jx − x0j > R). Here R is called radius of convergence. In many cases the radius of convergence can be found by using the following formulas, a 1 R = lim n or R = lim pn n!0 an+1 n!0 janj whenever the limits exist. Regardless of the existence of the above limits, it is known that R always exists. Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 7 / 48 Differentiation of power series: Suppose that the power series (0.1) converges for jxj < R with R > 0 and denote the sum by f (x): 1 X n 2 3 f (x) = anx = a0 + a1x + a2x + a3x + ··· : n=0 Then f (x) is automatically is continuous and has the derivatives of all orders for jxj < R. Also, 1 0 X n−1 2 f (x) = nanx = a1 + 2a2x + 3a3x + ··· ; (0.6) n=1 1 00 X n−2 f (x) = n(n − 1)x = 2a2 + 3 · 2a3x + ··· ; (0.7) n=2 and so on, and each of the resulting series converges for jxj < R. And we can link the coefficient an to f (x) and its derivative via the following formula f (n)(0) a = (0.8) n n! Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 8 / 48 P1 n P1 n Algebra of power series: Let f (x) = n=0 anx and g(x) = n=0 bnx be two power series with radius of convergence at least R > 0, then these power series can be added or subtracted termwise: 1 X n f (x) ± g(x) = (an ± bn)x = (a0 ± b0) + (a1 ± b1)x + ··· : n=0 They can also be multiplies as they were polynomials, in the sense that 1 X n f (x)g(x) = cnx n=0 where cn = a0bn + a1bn−1 + ··· + anb0. If f (x) = g(x) for jxj < R if and only if an = bn for all n i.e. If both series converges to the same function for jxj < R if and only if they have the same coefficients. Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 9 / 48 Power series representation of a function: Let f be a continuous function defined for jxj < R or( −R; R) which has derivatives of all orders in the interval( −R; R). Can f (x) be represented by a power series about the point zero? Equivalently will the following hold 1 X f (n)(0) f 00(0) f (x) = xn = f (0) + f 0(0)x + x2 + ··· (0.9) n! 2! n=0 throughout the interval( −R; R)? This is often true, but unfortunately it is some times false. The above expansion is valid if the error term Rn(x) in Taylor's formula: n X f (k)(0) f (x) = xk + R (x) k! n n=0 convegres to zero as n tends to infity. Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 10 / 48 By means of the procedure explained in the previous slide, it is quite easy to obtaion the following familiar expansions, 1 X xn x x2 x3 ex = = 1 + + + + ··· ; (0.10) n! 1! 2! 3! n=0 1 X x2n+1 x3 x5 sin x = (−1)n = x − + − · · · ; (0.11) (2n + 1)! 3! 5! n=0 1 X x2n x2 x4 cos x = (−1)2n = 1 − + − · · · ; (0.12) (2n)! 2! 4! n=0 1 1 X = (±1)nxn = 1 ± x + x2 ± x3 + ··· ; (0.13) 1 ± x n=0 1 X xn x2 x3 x4 log (1 + x) = (−1)n−1 = x − + − + ··· ; (0.14) n 2 3 4 n=1 1 X x2n+1 x3 x5 tan−1 x = (−1)n = x − + − · · · ; (0.15) (2n + 1) 3 5 n=0 Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 11 / 48 The function f for which the above series expansion (0.9) is valid for some neighbourhood of zero is said to be analytic at x = 0: More generally the analyticity at any point is defined as follows. Analytic at a point: A function f (x) with the property that a power series expansion of the form 1 X n f (x) = an(x − x0) n=0 valid in some neighbourhood of the point x0 is said to be analytic at x0. In this case the coefficients an's are necessarily given by f (n)(x ) a = 0 ; n n! and the above series is called the Taylor series of f (x) at x0. Pradeep Boggarapu (Dept. of Maths) Review of Power Series September 16, 2015 12 / 48 Facts about analytic functions: Polynomials and the functions ex , sin x and cos x are analytic at all points. If f (x) and g(x) are analytic at x0, then f (x) + g(x), f (x)g(x), and f (x)=g(x) [if g(x0) 6= 0] are also analytic at x0 .
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