Mathematical functions 1.2.4



	You are here:		Chapter: 1 Units and fundamental constants Section: 1.2 Fundamental physical constants SubSection: 1.2.4 Mathematical functions

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1.2.4 Mathematical functions

The main purpose of this book is to tabulate values of physical and chemical constants. Previous editions included, in addition, short tables of logarithms, trigonometrical and other elementary mathematical functions as an aid to the calculation of experimental results. The widespread use of pocket calculators and of computers has now made these unnecessary. However, although the equations of mathematical physics can now readily be solved directly in numerical terms, there remain occasions when it is helpful to express the solutions using familiar tabulated functions; we therefore give references to tables where values of these functions can be found.

Exponential and trigonometric functions appear as the solutions of linear differential equations with constant coefficients. Many other functions appear in the solution of differential equations with stated boundary conditions, particularly in the determination of fields (electromagnetic, hydrodynamic, or gravitational, for example). Bessel functions occur in the solution of potential problems in two dimensions, or in three dimensions with cylindrical symmetry, and in a great variety of other problems. In three-dimensional potential problems involving spheres or spheroids—the Earth’s gravitational and magnetic fields, for example—it is convenient to work in terms of surface harmonics and Legendre functions, as in sections 2.7.5 and 2.7.6. Elliptic functions and related integrals occur in the solution of certain non-linear differential equations. The references below include tables of these functions and of elementary functions to numbers of decimal places beyond those usually available on pocket calculators, and also tables for group theory which finds extensive applications in many areas of physics and chemistry.

Table of numerical constants Note: all values are approximate.
π	3.14159	26535	898
π² . . . . . . . .	9.86960	44010	894
ln(π) . . . . . . .	1.14472	98858	494
log(π) . . . . . . .	0.49714	98726	941
e . . . . . . . .	2.71828	18284	590
e² . . . . . . . .	7.38905	60989	307
log(e) . . . . . . .	0.43429	44819	033
e^π . . . . . . . .	23.14069	26327	793
e^−π . . . . . . . .	0.04321	39182	637(72)

. . . . . . .	1.41421	35623	731
. . . . . . . .	1.73205	08075	689
. . . . . . . .	2.23606	79774	998
. . . . . . . .	3.16227	76601	684
. . . . . . . .	1.25992	10498	949

ln(2) . . . . . . .	0.69314	71805	599
ln(3) . . . . . . .	1.09861	22886	681
ln(10) . . . . . . .	2.30258	50929	940

deg/rad . . . . . . .	57.29577	95130	823
rad/deg . . . . . . .	0.01745	32925	199(43)
Euler’s constant γ . . . .	0.57721	56649	015

References

M. Abramowitz and I. A. Stegun (eds) (1965) Handbook of Mathematical Functions with Formulas, Graphs and    Mathematical Tables, NBS Applied Mathematics Series No. 55. Includes extensive tables of functions to many decimal places, e.g. exponentials to 15 places, sines and cosines to 23 places.
P. W. Atkins, M. S. Child and C. S. G. Phillips (1970) Tables for Group Theory, Oxford University Press.
National Bureau of Standards (Federal Works Agency, Works Projects Administration of the City of New York)
   Tables of the Exponential Function (1939) (12–18 decimal places).
   Tables of Sines and Cosines (1940) (radian arguments, 8 places, increasing to 15 for small arguments).
   Tables of Sine, Cosine and Exponential Integrals (1940) (9 places).
Jahnke and Emde (revised by Lösch) (1960) Tables of Higher Functions, McGraw-Hill. Includes, among others, gamma function, exponential, sine and cosine integrals, elliptic functions, orthogonal polynomials, Legendre functions, Bessel functions, Mathieu functions and confluent hypergeometric functions.

A.E. Bailey

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