A necessary and sufficient condition for a function f x to be represented by its Taylor series in a neighborhood of a point x0 is that the remainder term in the Taylor formula* should te
Trang 1A special case of the Taylor series (for x0=0) is the Maclaurin series:
∞
n=0
1
n!f
(n)(0)x n = f (0) + f (0)x + 1
2f (0)x2+· · ·
A formal Taylor series (Maclaurin series) for a function f (x) may be:
1) divergent for x≠x0,
2) convergent in a neighborhood of x0to a function different from f (x),
3) convergent in a neighborhood of x0to the function f (x).
In the last case, one says that f (x) admits expansion in Taylor series in the said neighborhood,
and one writes
f (x) =
∞
n=0
1
n!f (n)(x
0)(x – x0)n.
8.3.2-2 Conditions of expansion in Taylor series
A necessary and sufficient condition for a function f (x) to be represented by its Taylor
series in a neighborhood of a point x0 is that the remainder term in the Taylor formula*
should tend to zero as n → ∞ in this neighborhood of x0.
In order that f (x) could be represented by its Taylor series in a neighborhood of x0,
it suffices that all its derivatives in that neighborhood be bounded by the same constant,
|f(n) (x)| ≤M for all n.
Uniqueness of the Taylor series expansion For a function f (x) that can be represented
as the sum of a power series, the coefficients of this series are determined uniquely (since
this series is the Taylor series of f (x) and its coefficients have the form f
(n) (x0)
n! , where
n=0, 1, 2, ) Therefore, in problems of representing a function by a power series, the
answer does not depend on the method adopted for this purpose
8.3.2-3 Representation of some functions by the Maclaurin series
The following representations of elementary functions by Maclaurin series are often used
in applications:
e x=1+ x + x
2
2! +
x3
3! +· · · + x n
n! +· · · ;
sin x = x – x
3
3! +
x5
5! –· · · + (–1)n–1 x
2n–1 (2n–1)! +· · · ;
cos x =1– x
2
2! +
x4
4! –· · · + (–1)n x
2n
(2n)! +· · · ;
sinh x = x + x3
3! +
x5
5! +· · · + x2n–1
(2n–1)! +· · · ;
* Different representations of the remainder in the Taylor formula are given in Paragraph 6.2.4-4.
Trang 2354 SERIES
cosh x =1+ x
2
2! +
x4
4! +· · · + x2n
(2n)! +· · · ;
(1+ x) α =1+ αx + α (α –1)
2! x
2+· · · + α (α –1) (α – n +1)
n+· · · ;
ln(1+ x) = x – x
2
2 +
x3
3 –· · · + (–1)n+1
x n
n +· · · ;
arctan x = x – x3
3 +
x5
5 –· · · + (–1)n+1
x2n–1
2n–1 +· · ·
The first five series are convergent for –∞ < x < ∞ (R = ∞), and the other series have unit
radius of convergence, R =1
8.3.3 Operations with Power Series Summation Formulas for
Power Series
8.3.3-1 Addition, subtraction, multiplication, and division of power series
1 Addition and subtraction of power series Two series ∞
n=0a n x
n and ∞
n=0b n x
n with convergence radii R a and R b, respectively, admit term-by-term addition and subtraction on the intersection of their convergence intervals:
∞
n=0
a n x n
∞
n=0
b n x n=
∞
n=0
c n x n, c n = a n b n.
The radius of convergence of the resulting series satisfies the inequality R c≥min[R a , R b]
2 Multiplication of power series Two series∞
n=0a n x
nand∞
n=0b n x
n, with the respective convergence radii R a and R b, can be multiplied on the intersection of their convergence intervals, and their product has the form
∞ n=0
a n x n∞
n=0
b n x n
=
∞
n=0
c n x n, c
n=
n
k=0
a k b n–k.
The convergence radius of the product satisfies the inequality R c≥min[R a , R b]
3 Division of power series The ratio of two power series∞
n=0a n x
nand∞
n=0b n x
n , b
0≠ 0,
with convergence radii R a and R bcan be represented as a power series
∞
n=0a n x
n
∞
n=0b n x
n
= c0+ c1x + c2x2+· · · =∞
n=0
c n x n, (8.3.3.1)
whose coefficients can be found, by the method of indefinite coefficients, from the relation
(a0+ a1x + a2x2+· · ·) = (b0+ b1x + b2x2+· · ·)(c0+ c1x + c2x2+· · ·).
Trang 3Thus, for the unknown c n, we obtain a triangular system of linear algebraic equations
a n=
n
k=0
b k c n–k, n=0, 1, ,
which is solved consecutively, starting from the first equation:
c0= a0
b0, c1=
a1b0– a0b1
b2 0
, c n = a b n
0 –
1
b0
n
k=1
b k c n–k, n=2, 3,
The convergence radius of the series (8.3.3.1) is determined by the formula
R1= min
*
R a,M ρ+1
+
,
where ρ is any constant such that0< ρ < R b ; ρ can be chosen arbitrarily close to R b ; and M
is the least upper bound of the quantities|b m /b0|ρ m (m =1,2, ), so that|b m /b0|ρ m≤M
for all m.
8.3.3-2 Composition of functions representable by power series
Consider a power series
z = f (y) = a0+ a1y + a2y2+· · · =∞
n=0
a n y n (8.3.3.2)
with convergence radius R Let the variable y be a function of x that can be represented by
a power series
y = ϕ(x) = b0+ b1x + a2x2+· · · =
∞
n=0
b n x n (8.3.3.3)
with convergence radius r It is required to represent z as a power series of x and find the
convergence radius of this series
Formal substitution of (8.3.3.3) into (8.3.3.2) yields
z = f ϕ (x)
=
∞
n=0
a n
∞ k=0
b k x kn
= A0+ A1x + A2x2+· · · =∞
n=0
A n x n, (8.3.3.4)
where
A0 = a0+ a1b0+ a2b2
0+· · · ,
A1 = a1b1+2a2b0b1+3a3b2
0b1+· · · ,
A2 = a1b2+ a2(b21+2b0b2) +3a3(b0b2
1+ b20b2) +· · · ,
Trang 4356 SERIES
THEOREM ON CONVERGENCE OF SERIES(8.3.3.4)
(i) If series (8.3.3.2) is convergent for all y (i.e., R = ∞), then the convergence radius
of series (8.3.3.4) coincides with the convergence radius r of series (8.3.3.3).
(ii) If 0 ≤ |b0|< R, then series (8.3.3.4) is convergent on the interval (–R1, R1), where
R1 = (R –|b0|)ρ
M + R –|b0|,
and ρ is an arbitrary constant such that0< ρ < r; ρ can be chosen arbitrarily close to r; and
Mis the least upper bound of the quantities|b m|ρ m (m =1, 2, ), so that|b m|ρ m≤M for
all m.
(iii) If |b0|> R, then series (8.3.3.4) is divergent.
Remark Case (i) is realized, for instance if (8.3.3.2) has finitely many terms.
8.3.3-3 Local inversion of a function represented by power series
1 Suppose that y = y(x) is a function that can be represented, in a neighborhood of a point x = x0, by the power series
y = y0+ a(x – x0) + b(x – x0)2+ c(x – x0)3+ d(x – x0)4+· · · , a≠ 0
Then the inverse function x = x(y), in a neighborhood of y = y0, can be represented by the series
x = x0+ 1
a (y – y0) – b
a3(y – y0)2+
2b2– ac
a5 (y – y0)3+
5abc–5b3– a2d
a7 (y – y0)4+· · ·
2 B¨urman–Lagrange formula Suppose that for a given function
y = f (x), (8.3.3.5) the auxiliary function
ϕ (x) = x
f (x)
is holomorphic in a neighborhood of the point x = 0 (i.e., it can be represented by a
convergent power series in a neighborhood of that point) Then there is ε >0such that on the interval|y|< ε, the function (8.3.3.5) is invertible and its inverse x = g(y) is holomorphic
on that interval,
x=
∞
n=1
b n y n, b
n= 1
n!
d n–1
dx n–1ϕ n (x)
x=0.
The expression for the coefficients b n is called the B¨urman–Lagrange formula.
Example 1 Consider the function
y = x(x + b) (b≠ 0 ),
for which the auxiliary function has the form ϕ(x) = (x + a)– 1 Using the B¨urman–Lagrange formula and the relation
d n–1
dx n–1
1
(x + a) n = (–1 )n–1n (n +1 )· · · (2n– 2 )
(x + a)2n–1 ,
we find the representation of the given function by power series:
x= y
a –y
2
a3 +· · · + (–1 )n–1 (2n– 2 )!
(n –1)! n!
y
a2n–1 +· · ·
Trang 58.3.3-4 Simplest summation formulas for power series.
Suppose that the sum of a power series is known,
∞
k=0
a k x k = S(x). (8.3.3.6)
Then, using term-by-term integration (on the convergence interval), one can find the fol-lowing sums:
∞
k=0
a k k m x k=
x d dx
m
S (x);
∞
k=0
a k (nk + m)x nk+m–1 = dx d
x m S (x n)
;
∞
k=0
a k
nk + m x
nk+m= x
0 x
m–1S (x n ) dx, n>0, m >0;
∞
k=0
a k nk nk + m + s x nk+s = x dx d
x s–m x
0 x
m–1S (x n ) dx
, n>0, m >0;
∞
k=0
a k nk nk + m + s x nk+s =
x
0 x
s–m d
dx
*
x m S (x n)+
dx, n>0, s >0
(8.3.3.7)
Example 2 Let us find the sum of the series∞
n=0kx
k–1
We start with the well-known formula for the sum of an infinite geometrical progression:
∞
k=0
x k= 1
1– x (|x| < 1 ).
This series is a special case of (8.3.3.6) with a k= 1, S(x) =1/( 1– x) The series∞
n=0kx
k–1 can be obtained
from the left-hand side of the second formula in (8.3.3.7) for m =0and n =1 Substituting S(x) =1/( 1– x)
into the right-hand side of that formula, we get
∞
k=0
kx k–1= d
dx
1
1– x =
1
( 1– x)2 (|x| < 1 ).
8.4 Fourier Series
8.4.1 Representation of 2π-Periodic Functions by Fourier Series.
Main Results
8.4.1-1 Dirichlet theorem on representation of a function by Fourier series
A function f (x) is said to satisfy the Dirichlet conditions on an interval (a, b) if:
1) this interval can be divided into finitely many intervals on which f (x) is monotone
and continuous;
2) at any discontinuity point x0 of the function, there exist finite one-sided limits
f (x0+0) and f (x0–0)
Trang 6358 SERIES
DIRICHLET THEOREM Any2π-periodic function that satisfies the Dirichlet conditions
on the interval (–π, π) can be represented by its Fourier series
f (x) = a0
2 +
∞
n=1
a n cos nx + b n sin nx
(8.4.1.1)
whose coefficients are defined by the Euler–Fourier formulas
a n= 1
π
π
–π f (x) cos nx dx, n=0, 1, 2, ,
b n= 1
π
π
–πf (x) sin nx dx, n=1, 2, 3,
(8.4.1.2)
At the points of continuity of f (x), the Fourier series converges to f (x), and at any discontinuity point x0, the series converges to 12[f (x0+0) + f (x0–0)]
The coefficients a n and b n of the series (8.4.1.1) are called the Fourier coefficients.
Remark. Instead of the integration limits –π and π in (8.4.1.2), one can take c and c +2π , where c is an
arbitrary constant.
8.4.1-2 Lipschitz and Dirichlet–Jordan convergence criteria for Fourier series
LIPSCHITZ CRITERION.Suppose that f (x) is continuous at a point x0and for sufficiently
small ε >0satisfies the inequality |f (x0 ε ) – f (x0)| ≤Kε σ , where L and σ are constants,
0< σ≤ 1 Then the representation (8.4.1.1)–(8.4.1.2) holds at x = x0
In particular, the conditions of the Lipschitz criterion hold for continuous piecewise differentiable functions
Remark The Fourier series of a continuous periodic function with no additional conditions (for instance,
of its regularity) may happen to be divergent at infinitely many (even uncountably many) points.
DIRICHLET–JORDAN CRITERION Suppose that f (x) is a function of bounded variation
on some interval (x0 – h, x0+ h)(–π, π) (i.e., f (x) can be represented as a difference of
two monotonically increasing functions) Then the Fourier series (8.4.1.1)–(8.4.1.2) of the
function f (x) at the point x0converges to the value 12[f (x0+0) + f (x0–0)]
8.4.1-3 Riemann localization principle
RIEMANN LOCALIZATION PRINCIPLE The behavior of the Fourier series of a function
f (x) at a point x0* depends only on its values near that point, i.e., values in an arbitrarily small neighborhood of that point
Thus, for two functions that coincide in a neighborhood of a point x0, but differ
outside that neighborhood, the corresponding Fourier series at x0are either both convergent
or divergent and have the same sum in the case of convergence, although their Fourier coefficients may be different, being dependent on all values of the functions
* What is meant here is the fact of convergence or divergence of the Fourier series at x0, and also the numerical value of its sum in the case of convergence.
Trang 78.4.1-4 Asymptotic properties of Fourier coefficients.
1◦ Fourier coefficients of an absolutely integrable function tend to zero as n goes to infinity:
a n →0and b n →0as n → ∞.
2◦ Fourier coefficients of a continuous 2π-periodic function have the following limit properties:
lim
n→∞ (na n) =0, lim
n→∞ (nb n) =0,
i.e., a n = o(1/n ), b n = o(1/n)
3◦ If a continuous periodic function is continuously differentiable up to the order m –1
inclusively, then its Fourier coefficients have the following limit properties:
lim
n→∞ (n m a n) =0, lim
n→∞ (n m b n) =0,
i.e., a n = o n–m
, b n = o n–m
8.4.1-5 Integration and differentiation of Fourier series
1◦ The Fourier series of a continuous periodic function of bounded variation admits
term-by-term integration, and the resulting series is uniformly convergent
2◦ The Fourier series of a k times continuously differentiable function admits term-by-term differentiation (k –1) times, the resulting series still being uniformly convergent (the
k th differentiation yields the kth derivative of the function, but the resulting series may have
only mean-square convergence, not necessarily pointwise convergence)
8.4.2 Fourier Expansions of Periodic, Nonperiodic, Odd, and Even
Functions
8.4.2-1 Expansion of2l-periodic and nonperiodic functions in Fourier series
1◦ The case of2l-periodic functions can be easily reduced to that of2π-periodic functions
by changing the variable x to z = πx
l In this way, all the results described above for
2π-periodic functions can be easily extended to2l-periodic functions
The Fourier expansion of a2l -periodic function f (x) has the form
f (x) = a0
2 +
∞
n=1
a ncosnπx l + b nsinnπx l
, (8.4.2.1) where
a n= 1
l
l
–lf (x) cos
nπx
l dx, b n= 1
l
l
–lf (x) sin
nπx
l dx (8.4.2.2)
2◦ A nonperiodic (aperiodic) function f (x) defined on the interval (–l, l) can also be
represented by a Fourier series (8.4.2.1)–(8.4.2.2); however, outside that interval, the sum
of that series S(x) may differ from f (x)*.
* The sum S(x) is a2l -periodic function defined for all x, but f (x) may happen to be nonperiodic, or even undefined outside the interval (–l, l).