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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

eπ     .    .    .    .    .    .    .    .

23.14069

26327

793

eπ   .    .    .    .    .    .    .    .

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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