POWER SERIES IN TWO VARIABLES
2. POWER SERIES IN TWO VARIABLES
Just as in the one-variable case discussed in Chapter 1, the power series (about a given point) for a function of two variables is unique, and we may use any convenient method of finding it (see Chapter 1 for methods).
Example 1. Expand f (x, y) = sin x cos y in a two-variable Maclaurin series. We write and multiply the series for sin x and cos y. This gives
3! 2! Example 2. Find the two-variable Maclaurin series for ln(1 + x − y). We replace x in
equation (13.4) of Chapter 1 by x − y to get
ln(1 + x − y) = (x − y) − (x − y) 2 /2 + (x − y) 3 /3 + · · ·
2 2 /2 + x 3 =x−y−x 2 /2 + xy − y /3 − x y + xy 2 −y 3 /3 + · · · . The methods of Chapter 1, used as we have just shown, provide an easy way
of obtaining the power series for many simple functions f (x, y). However, it is also convenient, for theoretical purposes, to have formulas for the coefficients in the Taylor series or the Maclaurin series for f (x, y); see, for example, Problem 8.2. Following a process similar to that used in Chapter 1, Section 12, we can find the coefficients of the power series for a function of two variables f (x, y) (assuming that it can be expanded in a power series). To find the series expansion of f (x, y) about the point (a, b) we write f (x, y) as a series of powers of (x − a) and (y − b) and then differentiate this equation repeatedly as follows.
10 (x − a) + a 01 (y − b) + a 20 (x − a) +a 11 (x − a)(y − b) +a
f (x, y) = a 00 +a
02 (y − b) +a 30 (x − a) +a 21 (x − a) (y − b) +a
12 (x − a)(y − b) +a 03 (y − b) +···.
f xx = 2a 20 + terms containing (x − a) and/or (y − b),
f xy =a 11 + terms containing (x − a) and/or (y − b). [We have written only a few derivatives to show the idea. You should be able to
calculate others in the same way (Problem 7).] Now putting x = a, y = b in (2.1),
192 Partial Differentiation Chapter 4
we get
f (a, b) = a 00 , f x (a, b) = a 10 , f y (a, b) = a 01 , (2.2)
f xx (a, b) = 2a 20 , f xy (a, b) = a 11 , etc.
[Remember that f x (a, b) means that we are to find the partial derivative of f with respect to x and then put x = a, y = b, and similarly for the other derivatives.] Substituting the values for the coefficients into (2.1), we find
f (x, y) = f (a, b) + f x (a, b)(x − a) + f y (a, b)(y − b)
1 + [f xx
(a, b)(x − a)(y − b) + f 2 (a, b)(y − b) ]···. 2!
(a, b)(x − a) 2 + 2f xy
yy
This can be written in a simpler form if we put x − a = h and y − b = k. Then the second-order terms (for example) become
[f xx (a, b)h 2 + 2f
xy (a, b)hk + f yy (a, b)k ].
We can write this in the form
if we understand that the parenthesis is to be squared and then a term of the form h(∂/∂x)k(∂/∂y)f (a, b) is to mean hkf xy (a, b). It can be shown (Problem 7) that the third-order terms can be written in this notation as
and so on for terms of any order. Thus we can write the series (2.3) in the form
The numbers appearing in the nth order terms are the familiar binomial coefficients [in the expansion of (p + q) n ] divided by (n!). (See Chapter 1, Section 13C.)
PROBLEMS, SECTION 2
Find the two-variable Maclaurin series for the following functions. 1. cos x sinh y
2. cos(x + y)
3. ln(1 + x) 1+y
6. e x+y 7. Verify the coefficients of the third-order terms [(2.6) or n = 3 in (2.7)] of the power
4. e xy
5. p1 + xy
series for f (x, y) by finding the third-order partial derivatives in (2.1) and substi- tuting x = a, y = b.
8. Find the two-variable Maclaurin series for e x cos y and e x sin y by finding the series for e z =e x+iy and taking real and imaginary parts. (See Chapter 2.)
Section 3 Total Differentials 193
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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