APPENDX E
INTERGRALS, SUMS, INFINITE SERIES AND PRODUCTS, AND CONTINUED FRACTIONS
Integral Tables
E-l. Elementary Indefinite Integrals
Integrals containing |
Start with formula number |
Algebraic functions |
1 |
Trigonometric functions |
9 |
Exponential and hyperbolic functions |
38 |
Logarithmic functions |
61 |
Sums and Infinite Series
E-5. Miscellaneous Infinite Series
E-6. Power Series: Binomial Series
E-7. Power Series for Elementary Transcendental Functions
Infinite Products and Continued Fractions
E-1. Elementary Indefinite Integrals. Add constant of integration in each case.
E-2. Indefinite Integrals.* Add constants of integration as needed. Note: As customary in integral tables, interpret loge f(x) as loge |f(x)| whenever it occurs on the right-hand side of an integral formula and f(x) is negative, m, n are integers.
(a) Integrals containing ax + b (a≠0)
* Adapted, in part, from I. Bronstein and K. Semendjajev, Pocketbook of Mathematics, 6th ed., published by the Soviet Government, Moscow, 1956.
(b) Integrals containing ax + b and cx + d (a≠0, c≠0)
(c) Integrals containing a + x and b + x (a≠b)
(d) Integrals containing ax2 + bx + c (a≠0)
(e) Integrals containing ax2 ± x2
Where ± or 
appears in a formula, the upper sign refers to X ≡ a2 + x2 and the lower sign to X ≡ a2 – x2 (a ≠ 0).
(f) Integrals containing a3 ± x2, with
(g) Integrals containing a4 ± x4 (a ≠ 0)
(h) Integrals containing 
and ax2 + b2x (a, b ≠0)
(i) Integrals containing and ax2 − b2x > 0 (a, b ≠ 0)
(j) Other integrals containing 
(k) Integrals containing 
(l) Integrals containing 
with
(m) Integrals containing 
with
(n) Integrals containing 
with
(o) Integrals containing 
with
(p) Other irrational forms (a > 0, b ± 0)
(q) Recursion formulas (m, n, p are integers)
(r) Integrals containing the sine function (a ≠ 0)
(s) Integrals containing the cosine function (a ≠ 0)
(t) Integrals containing both sine and cosine (a ≠ 0)
(u) Integrals containing tangent and cotangent functions (a ≠ 0)
(v) Integrals containing hyperbolic functions (a ≠ 0)
(w) Integrals containing exponential functions
(x) Integrals containing logarithmic functions
(y) Integrals containing inverse trignometric and hyperbolic functions
E.3 Definite Integrals (see also Secs. 21.4-1 and 21.6-6 and Appendix D). m,n are integers
(a) Integrals containing algebraic functions
(b) Integrals containing trigonometric functions (see also Secs. E-Sc and d)
(c) Integrals containing exponential and hyperbolic functions (a > 0)
(d) Integrals containing logarithmic functions (see also Secs. E-Sc and d)
SUMS AND INFINITE SERIES
E-5. Miscellaneous Infinite Series
NOTE: The series 
is referred to as Riemann's zeta function of the variable z.
E-6. Power Series: Binomial Series
NOTE: The series for (1± x)m reduces to finite sums (Binomial Theorem, Sec. 1.4-1) if m = 1, 2, 3, . . . .
E-7. Power Series for Elementary Transcendental Functions
where the Bk are the Bernoulli numbers defined in Sec. 21.5-2. 13 and 14 yield similar series for tanh z and coth z with the aid of Eq. (21.2-32).
INFINITE PRODUCTS AND CONTINUED FRACTIONS
E-8. Some Infinite Products (see also Sec. 7-6)
E-9. Some Continued Franctions (see also Secs. 4.8-8 and 21.5-7)
Table E-1. Operations with Series*
BIBLIOGRAPHY
Abramowitz, M., and I. A. Stegun: Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.
Bierens de Haan, D.: Nouvelles tables d' intégrates definies, Stechert, New York, 1939.
Byrd, P. F., and M. D. Friedman: Handbook of Elliptic Integrals for Engineers and Physicists, Springer, Berlin, 1954.
Gröbner and Hofreiter: Integral Tafel, 2d ed., Springer, Vienna, 1958.
Lindman, C. F.: Examen des novelles tables d' integrates definies de M. Bierens de Haan, 1944.
Luke, Y. I.: Integrals of BesselFunctions, McGraw-Hill, New York, 1962.
Petit Bois, G.: Tables of Indefinite Integrals, Dover, New York, 1961.
Ryshik, I. M., and I. S. Gradstein: Tables of Series, Products, and Integrals, Academic Press, New York, 1964.