Handbook of mathematics for engineers and scienteists part 139 potx

Proceeding in this way, we can eliminate all functions f i t and obtain a functional-differential equation with two variables of the form 17.5.5.10–17.5.5.11, which can be reduced to the

934 DIFFERENCEEQUATIONS ANDOTHERFUNCTIONALEQUATIONS

where B1, , B4 are arbitrary constants and k1 and k2 are the roots of the quadratic equation

(k – A1)(k – A4) – A2A3=0 (17.5.5.23)

In the degenerate case k1 = k2, the terms e k2ϕ and e–k2z in (17.5.5.22) should be replaced with ϕe k1ϕ and ze–k1z, respectively In the case of purely imaginary or complex roots, one

should separate the real (or imaginary) part of the roots in solution (17.5.5.22)

On substituting (17.5.5.22) into the original functional equation, one obtains conditions

for the free coefficients and identifies the function f (t), namely,

B2= B4=0 =⇒ f(t) = [A2A3+ (k1– A1)2]B1B3e–k1ψ,

B1= B3=0 =⇒ f(t) = [A2A3+ (k2– A1)2]B2B4e–k2ψ,

A1=0 =⇒ f(t) = (A2A3+ k21)B1B3e–k1ψ + (A

2A3+ k22)B2B4e–k2ψ.

(17.5.5.24)

Solution (17.5.5.22), (17.5.5.24) involves arbitrary functions ϕ = ϕ(x) and ψ = ψ(t).

Degenerate case In addition, the functional equation has two degenerate solutions

[formulas (17.5.5.7) are used],

f = B1B2e A1ψ, g = A

2B1e–A1ϕ, h = B

1e–A1ϕ, R = –B

2e A1z – A

2Q,

where ϕ = ϕ(x), ψ = ψ(t), and Q = Q(z) are arbitrary functions; A1, A2, B1, and B2 are arbitrary constants; and

f = B1B2e A1ψ, h = –B

1e–A1ϕ – A

2g, Q = A2B2e A1z, R = B

2e A1z, where ϕ = ϕ(x), ψ = ψ(t), and g = g(x) are arbitrary functions; and A1, A2, B1, and B2are arbitrary constants

3◦ Consider a more general functional equation of the form

f (t) + g1(x)Q1(z) + · · · + g n (x)Q n (z) =0, where z = ϕ(x) + ψ(t). (17.5.5.25)

By differentiation in x, this equation can be reduced to a functional differential equation,

which may be regarded as a bilinear functional equation of the form (17.5.5.8) Using formulas (17.5.5.9) for the construction of its solution, one can first obtain a system of ODEs and then find solutions of the original equation (17.5.5.25)

4◦ Consider a functional equation of the form

f1(t)g1(x) + · · · + f m (t)g m (x) + h1(x)Q1(z) + · · · + h n (x)Q n (z) =0, z = ϕ(x) + ψ(t).

(17.5.5.26)

Assume that g m (x)  0 Dividing equation (17.5.5.26) by g m (x) and differentiating the result in x, we come to an equation of the form

f1(t) ¯g1(x) + · · · + f m–1(t) ¯g m–1(x) +

2n



i=1

s i (x)R i (z) =0

with a smaller number of functions f i (t) Proceeding in this way, we can eliminate all functions f i (t) and obtain a functional-differential equation with two variables of the form

(17.5.5.10)–(17.5.5.11), which can be reduced to the standard bilinear functional equation

by the method of splitting

R EFERENCES FOR C HAPTER 17 935

17.5.5-4 Nonlinear equations containing the complex argument z = ϕ(t)θ(x) + ψ(t).

Consider a functional equation of the form

[α1(t)θ(x) + β1(t)]R1(z) + · · · + [α n (t)θ(x) + β n (t)]R n (z) =0, z = ϕ(t)θ(x) + ψ(t).

(17.5.5.27)

Passing in (17.5.5.27) from the variables x and t to new variables z and t [the function θ

is replaced by (z – ψ)/ϕ], we come to the bilinear equation of the form (17.5.5.8):

n



i=1

α i (t)zR i (z) +

n



i=1

[ϕ(t)β i (t) – ψ(t)α i (t)]R i (z) =0

Remark. Instead of the expressions α i (t)θ(x) + β i (t) in (17.5.5.27) linearly depending on the function

θ (x), one can consider polynomials of θ(x) with coefficients depending on t.

 See also Section T12.3 for exact solutions of some linear and nonlinear difference and

functional equations with several independent variables

References for Chapter 17

Acz´el, J., Functional Equations: History, Applications and Theory, Kluwer Academic, Dordrecht, 2002 Acz´el, J., Lectures on Functional Equations and Their Applications, Dover Publications, New York, 2006.

Acz´el, J., Some general methods in the theory of functional equations with a single variable New applications

of functional equations [in Russian], Uspekhi Mat Nauk, Vol 11, No 3 (69), pp 3–68, 1956.

Acz´el, J and Dhombres, J., Functional Equations in Several Variables, Cambridge University Press,

Cam-bridge, 1989.

Agarwal, R P., Difference Equations and Inequalities, 2nd Edition, Marcel Dekker, New York, 2000 Balasubrahmanyan, R and Lau, K., Functional Equations in Probability Theory, Academic Press, San

Diego, 1991.

Belitskii, G R and Tkachenko, V., One-Dimensional Functional Equations, Birkh¨auser Verlag, Boston, 2003 Castillo, E and Ruiz-Cobo, M R., Functional Equations and Modelling in Science and Engineering, Marcel

Dekker, New York, 1992.

Castillo, E and Ruiz-Cobo, R., Functional Equations in Applied Sciences, Elsevier, New York, 2005 Czerwik, S., Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co.,

Singapore, 2002.

Daroczy, Z and Pales, Z (Editors), Functional Equations — Results and Advances, Kluwer Academic,

Dordrecht, 2002.

Eichhorn, W., Functional Equations in Economics, Addison Wesley, Reading, Massachusetts, 1978 Elaydi, S., An Introduction to Difference Equations, 3rd Edition, Springer-Verlag, New York, 2005.

Goldberg, S., Introduction to Difference Equations, Dover Publications, New York, 1986.

Jarai, A., Regularity Properties of Functional Equations in Several Variables, Springer-Verlag, New York,

2005.

Kelley, W and Peterson, A., Difference Equations An Introduction with Applications, 2nd Edition, Academic

Press, New York, 2000.

Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific

Pub-lishers, Warsaw, 1985.

Kuczma, M., Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw, 1968 Kuczma, M., Choczewski, B., and Ger, R., Iterative Functional Equations, Cambridge University Press,

Cambridge, 1990.

Mickens, R., Difference Equations, 2nd Edition, CRC Press, Boca Raton, 1991.

Mirolyubov, A A and Soldatov, M A., Linear Homogeneous Difference Equations [in Russian], Nauka

Publishers, Moscow, 1981.

Mirolyubov, A A and Soldatov, M A., Linear Nonhomogeneous Difference Equations [in Russian], Nauka

Publishers, Moscow, 1986.

Nechepurenko, M I., Iterations of Real Functions and Functional Equations [in Russian], Institute of

Com-putational Mathematics and Mathematical Geophysics, Novosibirsk, 1997.

936 DIFFERENCEEQUATIONS ANDOTHERFUNCTIONALEQUATIONS

Pelyukh, G P and Sharkovskii, O M., Introduction to the Theory of Functional Equations [in Russian],

Naukova Dumka, Kiev, 1974.

Polyanin, A D., Functional Equations, From Website EqWorld — The World of Mathematical Equations,

http://eqworld.ipmnet.ru/en/solutions/fe.htm.

Polyanin, A D and Manzhirov, A V., Handbook of Integral Equations: Exact Solutions (Supplement Some

Functional Equations) [in Russian], Faktorial, Moscow, 1998.

Polyanin, A D and Zaitsev, V F., Handbook of Nonlinear Partial Differential Equations (Sections S.4

and S.5), Chapman & Hall/CRC Press, Boca Raton, 2004.

Polyanin, A D., Zaitsev, V F., and Zhurov, A I., Methods for the Solution of Nonlinear Equations of

Mathematical Physics and Mechanics (Chapter 16 Solution of Some Functional Equations) [in Russian],

Fizmatlit, Moscow, 2005.

Rassias, T M., Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000.

Samarskii, A A and Karamzin, Yu N., Difference Equations [in Russian], Nauka Publishers, Moscow, 1978 Samarskii, A A and Nikolaev, E S., Methods for the Solution of Grid Equations [in Russian], Nauka

Publishers, Moscow, 1972.

Sedaghat, H., Nonlinear Difference Equations Theory with Applications to Social Science Models, Kluwer

Academic, Dordrecht, 2003.

Sharkovsky, A N., Maistrenko, Yu L., and Romanenko, E Yu., Difference Equations and Their

Applica-tions, Kluwer Academic, Dordrecht, 1993.

Small, C G., Functional Equations and How to Solve Them, Springer-Verlag, Berlin, 2007.

Smital, J and Dravecky, J., On Functions and Functional Equations, Adam Hilger, Bristol-Philadelphia,

1988.

Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC Press,

Boca Raton, 2003.

Chapter 18

Special Functions and Their Properties

 Throughout Chapter 18 it is assumed that n is a positive integer unless otherwise

specified

18.1 Some Coefficients, Symbols, and Numbers

18.1.1 Binomial Coefficients

18.1.1-1 Definitions

C k

n=

n

k



k ! (n – k)!, where k=1, , n;

C0

a=1, C k

a =

a

k



= (–1)k (–a) k

k! =

a (a –1) (a – k +1)

k! , where k=1,2, Here a is an arbitrary real number.

18.1.1-2 Generalization Some properties

General case:

C b

a= Γ(a +1)

Γ(b +1)Γ(a – b +1), where Γ(x) is the gamma function.

Properties:

C0

a =1, C k

n=0 for k = –1, –2, or k > n,

C b+1

a = b+a1C a– b 1 = a b+– b1C a b, C a b + C a b+1= C a+ b+11,

C n

– 1 2= (–1)n

22n C2n n = (–1)n(2n–1)!!

(2n)!! ,

C n

1 2= (–1)n–1

n22n–1C2n– n–12= (–1)n–1

n

(2n–3)!!

(2n–2)!!,

C2n+1

n+1 2= (–1)n2– 4n–1C n

2n, C2n n+1 2 =2– 2n C2n

4n+1,

C1 2

n = 22n+1

πC n

2n

, C n/2

n = 22n

π C

(n–1 )/

937

938 SPECIALFUNCTIONS ANDTHEIRPROPERTIES

18.1.2 Pochhammer Symbol

18.1.2-1 Definition

(a) n = a(a +1) (a + n –1) = Γ(a + n)

Γ(a) = (–1)n

Γ(1– a)

Γ(1– a – n). 18.1.2-2 Some properties (k =1,2, ).

(a)0=1, (a) n+k = (a) n (a + n) k, (n) k= (n + k –1)!

(n –1)! ,

(a)–n= Γ(a – n)

Γ(a) =

(–1)n (1– a) n, where a≠ 1, , n;

(1)n = n!, (1/2)n=2– 2n(2n)!

n! , (3/2)n =2– 2n(2n+1)!

n! ,

(a + mk) nk = (a) mk+nk

(a) mk , (a + n) n=

(a)2n (a) n , (a + n) k=

(a) k (a + k) n (a) n .

18.1.3 Bernoulli Numbers

18.1.3-1 Definition

The Bernoulli numbers are defined by the recurrence relation

B0=1,

n–1



k=0

C k

n B k=0, n=2, 3,

Numerical values:

B0=1, B1 = –12, B2 = 16, B4= –301, B6= 421 , B8= –301 , B10= 665, .,

B2m+1 =0 for m=1,2,

All odd-numbered Bernoulli numbers but B1are zero; all even-numbered Bernoulli numbers have alternating signs

The Bernoulli numbers are the values of Bernoulli polynomials at x =0: B n = B n(0) 18.1.3-2 Generating function

Generating function:

x

e x–1 =

∞



n=0

B n x

n

n!, |x|<2π This relation may be regarded as a definition of the Bernoulli numbers

The following expansions may be used to calculate the Bernoulli numbers:

tan x =

∞



n=1

|B2n|22n(22n–1)

(2n)! x

2n, |x|< π

2;

cot x =

∞



n=0

(–1)n B2n 22n

(2n)!x

2n–1, |x|< π.

18.2 E RROR F UNCTIONS E XPONENTIAL AND L OGARITHMIC I NTEGRALS 939

18.1.4 Euler Numbers

18.1.4-1 Definition

The Euler numbers E nare defined by the recurrence relation

n



k=0

C2k

2n E2k=0 (even numbered),

E2n+1=0 (odd numbered),

where n =0, 1,

Numerical values:

E0 =1, E2= –1, E4 =5, E6= –61, E8=1385, E10= –50251, .,

E2n+1=0 for n=0, 1,

All Euler numbers are integer, the odd-numbered Euler numbers are zero, and the even-numbered Euler numbers have alternating signs

The Euler numbers are expressed via the values of Euler polynomials at x = 1/2:

E n=2n E

n(1/2), where n =0,1,

18.1.4-2 Generating function Integral representation

Generating function:

e x

e2x+1 =

∞



n=0

E n x

n

n!, |x|<2π This relation may be regarded as a definition of the Euler numbers

Representation via a definite integral:

E2n= (–1)n22n+1 ∞

0

t2n dt cosh(πt).

18.2 Error Functions Exponential and Logarithmic

Integrals

18.2.1 Error Function and Complementary Error Function

18.2.1-1 Integral representations

Definitions:

erf x = √2

π

 x

0 exp(–t

2) dt (error function, also called probability integral),

erfc x =1– erf x = √2

π

 ∞

x exp(–t

2) dt (complementary error function).

Properties:

erf(–x) = – erf x; erf(0) =0, erf(∞) =1; erfc(0) =1, erfc(∞) =0

940 SPECIALFUNCTIONS ANDTHEIRPROPERTIES

18.2.1-2 Expansions as x →0 and x → ∞ Definite integral.

Expansion of erf x into series in powers of x as x →0:

erf x = √2

π

∞



k=0

(–1)k x

2k+1

k! (2k+1) =

2

√

π exp –x2∞

k=0

2k x2k+1

(2k+1)!!.

Asymptotic expansion of erfc x as x → ∞:

erfc x = √1

π exp –x2M–1

m=0

(–1)m

1 2

m

x2m+1 + O |x|– 2M–1

, M =1,2,

0 erf t dt = x erf x –

1

2 +

1

2 exp(–x2).

18.2.2 Exponential Integral

18.2.2-1 Integral representations

Definition:

Ei(x) =

 x

–∞

e t

t dt= –

 ∞

–x

e–t

t dt for x<0,

Ei(x) = lim

ε→+0

 –ε

–∞

e t

t dt+

 x

ε

e t

t dt



for x>0 Other integral representations:

Ei(–x) = –e–x

 ∞ 0

x sin t + t cos t

x2+ t2 dt for x>0,

Ei(–x) = e–x

 ∞ 0

x sin t – t cos t

x2+ t2 dt for x<0,

Ei(–x) = –x

 ∞

1 e

–xt ln t dt for x>0,

Ei(x) = C + ln x +

 x 0

e t–1

t dt for x>0, whereC =0.5772 . is the Euler constant

18.2.2-2 Expansions as x →0 and x → ∞.

Expansion into series in powers of x as x →0:

Ei(x) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

C + ln(–x) +∞

k=1

x k

k ! k if x <0,

C + ln x +

∞



k=1

x k

k ! k if x >0

Asymptotic expansion as x → ∞:

Ei(–x) = e–x

n



k=1

(–1)k (k –1)!

x k + R n, R n<

n!

x n.