Mathematical Handbook for Scientists and Engineers

The following numerical tables are intended less for extensive numerical computations than as quantitative background material indicating the behavior of the most important transcendental functions.

The following numerical constants are frequently useful:

Table F-l. Table of Squares*

* This table is reprinted from Tabic XXVII of Fisher and Yates, Statistical Tables for Biological, Agricultural, and Medical Research, published by Oliver & Boyd, Ltd., Edinburgh, by permission of the authors and publishers.

Exact squares of four-figure numbers can be quickly calculated from the identity (a ± b)² = a² ± 2ab + b². Thus 693.3² = 480249 + 415.8 + 0.09 = 480664.89.

Table F-2. Five-place Common Logarithms of Numbers† 100-155

Table F-3. Natural Trigonometric Functions and Their Logarithms *

Table F-4. Values and Logarithms of Exponential and Hyperbolic Functions *

Table F-5. Natural, Napierian, or Hyperbolic Logarithms *

Table F-6a. Sine Integral Si(x)*

Table F-6b. S₁(x) and Cosine Integral Ci(x)*

Table F-7. Exponential and Related Integrals†

Table F-8. Complete Elliptic Integrals, K and E

Table F-9a. Factorials and Their Reciprocals*

Table F-9b. Coefficients of the Binomial Expansion*

Table F-10. Gamma and Factorial Functions: Γ(x) = (y)!*

Table F-ll. Bessel Functions: J₀(x) and J₁(x)*

Table F-ll. Bessel Functions: N₀(x) and N₁(x)

Table F-ll. Bessel Functions: I₀(x) and I₁(x)

Auxiliary Functions N₀(x) and N₁(x)for Small Values of Argument

For small values of the argument, N₀(x) and N₁(x)are rapidly changing functions and linear interpolation is inaccurate. These tables of auxiliary functions can be used to give accurate interpolated values. For values of the argument above 0.1 the main tables are satisfactory if interpolation formulas are used.

Auxiliary Functions K₀(x) and K₁(x)for Small Values of Argument

For small values of the argument, K₀(x) and K₁(x)are rapidly changing functions and linear interpolation is inaccurate. These tables of auxiliary functions can be used to give accurate interpolated values. For values of the argument above 0.1 the main tables are satisfactory if interpolation formulas are used.

Examples of use of auxiliary functions for small values of argument:

Example 1. N₀(0.115) = -0.0715 + 1.4610 X 1.0607 = -0.0715 - 1.4610+ 0.0887 = -1.4438.

Linear interpolation from the direct-reading table of No would give the less accurate value

N₀(0.115) = -1.4444

Example 2.

compared with the less accurate value of ? 5.648 obtained by linear interpolation of the table for N₁(x).

Auxiliary functions I₀(x),I₁(x), K₀(x),K₁(x)for large values of argument

Table F-12. Legendre Polynomials *

Table F-13. Probability Function or Error Integral: erf x *

Table F-14. Normal-distribution Areas *

Fractional parts of the total area (1.000) under the normal curve between the mean and a perpendicular erected at various numbers of standard deviations (x/σ) from the mean. To illustrate the use of the table, 39.065 per cent of the total area under the curve will lie between the mean and a perpendicular erected at a distance of 1.23σ from the mean.

Each figure in the body of the table is preceded by a decimal point.

Table F-15. Normal-curve Ordinates *

Ordinates (heights) of the unit normal curve. The height (y) at any number of standard deviations from the mean is

To obtain answers in units of particular problems, multiply these ordinates by where N is the number of cases, i the class interval, and σ the standard deviation.

Each figure in the body of the table is preceded by a decimal point.

Table F-16. Distribution of t *

Values of t corresponding to certain selected probabilities (i.e., tail areas under the curve). To illustrate: the probability is 0.05 that a sample with 20 degrees of freedom would have t = 2.086 or larger.

* This table is reproduced in abridged form from Table III of Fisher and Yates, Statistical Tables for Biological, Agricultural, and Medical Research, published by Oliver & Boyd, Ltd., Edinburgh, by permission of the authors and publishers.

Table F-17. Distribution of x² *

Values of x² corresponding to certain selected probabilities (i.e., tail areas under the curve). To illustrate: the probability is 0.05 that a sample with 20 degrees of freedom, taken from a normal distribution, would have x² = 31.410 or larger.

* This table is reproduced in abridged form from Table IV of Fisher and Yates Statistical Tables for Biological, Agricultural, and Medical Research, published bj Oliver & Boyd, Ltd., Edinburgh, by permission of the authors and publishers.

Table F-18. Distribution of F*

5 Per Cent (Roman Type) and 1 Per Cent (Boldface Type) Points for the Distribution of F

Table F-19. Random Numbers*

Table F-20. Normal Random Numbers

Table F-21. sin x/x*

Table F-22.* Chebyshev Polynomials T_n(x)

* From M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.

BIBLIOGRAPHY

Short Tables of Transcendental Functions

Abramowitz, M., and I. A. Stegun: Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.

Dwight, H. B.: Mathematical Tables, McGraw-Hill, New York, 1941.

Flugge, W.: Four-place Tables of Transcendental Functions, McGraw-Hill, New York, 1954.

Jahnke and F. Emde: Tables of Functions with Formulae and Curves, Dover, New York, 1954.

Statistical Tables

Beyer, W. H.: CRC Handbook of Tables for Probability and Statistics, Chemical Rubber Co., Cleveland, Ohio, 1966.

Burington, R. S., and D. C. May: Handbook of Probability and Statistics, 2d ed., McGraw-Hill, New York, 1967.

Hald, A.: Statistical Tables and Formulas, Wiley, New York, 1952.

Meredith, W.: Mathematical and Statistical Tables, McGraw-Hill, New York, 1967.

Owen, D. B.: Handbook of Statistical Tables, Addison-Wesley, Reading, Mass., 1962.

Pearson, E. S., and H. 0. Hartley: Biometrika Tables for Statisticians, Cambridge, New York, 1956.

Indices to Numerical Tables

Etherington, Harold (ed.): Nuclear Engineering Handbook, McGraw-Hill, New York, 1958.

Fletcher, A.: Guide to Tables of Elliptic Functions, Mathematical Tables and Other Aids to Computation, vol. 3, no. 24, 1948.

_____, J. C. P. Miller, and L. Rosenhead: Index of Mathematical Tables, Addison- Wesley, Reading, Mass., 1962.

Greenwood, J. A., and H. O Hartley: Guide to Tables in Mathematical Statistics, Princeton, Princeton, N.J., 1962.

* This table is reprinted from Table XXVII of Fisher and Yates, Statistical Tables for Biological, Agricultural, and Medical Research, Published by Oliver & Bond, Ltd., Edinburgh, by permission of the authors and publishers.