EXPANDING FUNCTIONS IN POWER SERIES
12. EXPANDING FUNCTIONS IN POWER SERIES
Very often in applied work, it is useful to find power series that represent given functions. We illustrate one method of obtaining such series by finding the series for sin x. In this method we assume that there is such a series (see Section 14 for discussion of this point) and set out to find what the coefficients in the series must
be. Thus we write (12.1)
sin x = a 0 +a 1 x+a 2 x 2 +···+a n x n +··· and try to find numerical values of the coefficients a n to make (12.1) an identity
(within the interval of convergence of the series). Since the interval of convergence of a power series contains the origin, (12.1) must hold when x = 0. If we substitute
x = 0 into (12.1), we get 0 = a 0 since sin 0 = 0 and all the terms except a 0 on the
24 Infinite Series, Power Series Chapter 1
right-hand side of the equation contain the factor x. Then to make (12.1) valid at x = 0, we must have a 0 = 0. Next we differentiate (12.1) term by term to get
cos x = a 1 + 2a 2 x + 3a 3 x 2 +···.
(This is justified by Theorem 1 of Section 11.) Again putting x = 0, we get 1 = a 1 . We differentiate again, and put x = 0 to get
− sin x = 2a 2 + 3 · 2a 3 x + 4 · 3a 4 x 2 +···, (12.3)
0 = 2a 2 .
Continuing the process of taking successive derivatives of (12.1) and putting x = 0, we get
− cos x = 3 · 2a 3 + 4 · 3 · 2a 4 x+···,
(12.4) sin x = 4 · 3 · 2 · a 4 + 5 · 4 · 3 · 2a 5 x+···,
0=a 4 ; cos x = 5 · 4 · 3 · 2a 5 +···,
1 = 5! a 5 ,···.
We substitute these values back into (12.1) and get
You can probably see how to write more terms of this series without further com- putation. The sin x series converges for all x; see Example 3, Section 10.
Series obtained in this way are called Maclaurin series or Taylor series about the origin. A Taylor series in general means a series of powers of (x − a), where a is some constant. It is found by writing (x − a) instead of x on the right-hand side of an equation like (12.1), differentiating just as we have done, but substituting x = a instead of x = 0 at each step. Let us carry out this process in general for a function
f (x). As above, we assume that there is a Taylor series for f (x), and write (12.6)
f (x) = a 0 +a 1 (x − a) + a 2 (x − a) 2 +a 3 (x − a) 3 +a 4 (x − a) 4 +··· +a
n (x − a) +···,
f ′ (x) = a 1 + 2a 2 (x − a) + 3a 3 (x − a) 2 + 4a 4 (x − a) 3 +··· + na n
(x − a) n−1 +···,
f ′′ (x) = 2a
2 + 3 · 2a 3 (x − a) + 4 · 3a 4 (x − a) +···
+ n(n − 1)(n − 2)a n−3
f (n) (x) = n(n − 1)(n − 2) · · · 1 · a n + terms containing powers of (x − a).
Section 13 Techniques for Obtaining Power Series Expansions
25 [The symbol f (n) (x) means the nth derivative of f (x).] We now put x = a in each
equation of (12.6) and obtain
f (a) = a ′′ 0 , f (a) = a 1 , f (a) = 2a 2 , (12.7)
f ′′′ (a) = 3! a
3 ,···,f (a) = n! a n .
(n)
[Remember that f ′′ (a) means to differentiate f (x) and then put x = a; f (a) means to find f ′′ (x) and then put x = a, and so on.]
We can then write the Taylor series for f (x) about x = a:
(a) + (x −a) f (x −a) n f (12.8) f (x) = f (a) + (x −a)f (n)
The Maclaurin series for f (x) is the Taylor series about the origin. Putting
a = 0 in (12.8), we obtain the Maclaurin series for f (x):
x n (12.9)
n! We have written this in general because it is sometimes convenient to have the
formulas for the coefficients. However, finding the higher order derivatives in (12.9) for any but the simplest functions is unnecessarily complicated (try it for, say, e tan x ). In Section 13, we shall discuss much easier ways of getting Maclaurin and Taylor series by combining a few basic series. Meanwhile, you should verify (Problem 1, below) the basic series (13.1) to (13.5) and memorize them.
PROBLEMS, SECTION 12
1. By the method used to obtain (12.5) [which is the series (13.1) below], verify each of the other series (13.2) to (13.5) below.
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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