The material presented in this book has been compiled for the convenience of the reader. The aim of the book is to provide a handy source of formulas, conversion factors and constants for everyday use. The formulas and tables are amended by examples in all of those cases where their use is not self­ explanatory. The material has been selected to be helpful whenever it is inconvenient or not possible to consult tables available at the library. Section 1 provides the fundamental tools of mathematics needed in all areas of the physical sciences. Section 2 summarizes the SI system, lists conversion factors and provides precise values of fundamental constants. Sections 3 and 4 review the basic terms of spectroscopy, atomic structure and wave mechan­ ics. These sections serve as a guide to the interpretation of modern literature. Section 5 is a resource for work in the laboratory. Data and formulas are given to be of assistance in the use of frequently encountered equipment such as vacuum systems and electronic devices. Material constants and other data are listed for information and as an aid for estimates or problem solving. The assistance of the Springer-Verlag during the various stages of the development of this book is gratefully acknowl­ edged. The authors like to thank Dr. F. L. Boschke for his many helpful suggestions. Helmut J. Fischbeck Kurt H. Fischbeck Contents Basic mathematical facts and figures ............................... .

1 Basic mathematical facts and figures.- 1.1 Signs and symbols.- 1.1.1 Arithmetic, algebra and analysis of real scalar quantities.- 1.1.2 Arithmetic, algebra and analysis of complex scalar quantities.- 1.1.3 Matrices.- 1.1.4 Algebra and analysis of vectors and other tensors.- 1.1.5 Greek alphabet.- 1.1.6 Roman numerals.- 1.2 Numerical constants.- 1.3 Exponentials and roots.- 1.4 Logarithm.- 1.5 Complex numbers.- 1.6 Factorial.- 1.7 Mensuration formulas.- 1.8 Solid angle.- 1.8.1 Solid angle subtended by apertures.- 1.9 Trigonometric functions.- 1.9.1 Range and sign of trigonometric functions.- 1.9.2 Values of trigonometric functions for multiples of 30° (?/6) and 45° (?/4).- 1.9.2.1 Values of sine and cosine for multiples of ?.- 1.9.3 Relations between trigonometric functions.- 1.9.3.1 Relations involving one angle.- 1.9.3.2 Sums and differences of angles.- 1.9.3.3 Multiples of one angle.- 1.9.3.4 Functions of half angles.- 1.9.3.5 Sums and differences of trigonometric functions.- 1.9.3.6 Products and powers of trigonometric functions.- 1.9.4 Exponential definition of trigonometric functions.- 1.9.5 Inverse trigonometric functions.- 1.9.5.1 Fundamental relations for principal values of inverse trigonometric functions.- 1.9.6 Series expansion of trigonometric functions.- 1.9.6.1 Series expansion of inverse trigonometric functions.- 1.9.7 Small angle approximations.- 1.9.8 Values of trigonometric functions.- 1.10 Relations between sides and angles of a triangle.- 1.11 Hyperbolic functions.- 1.11.1 Relations between hyperbolic functions.- 1.11.1.1 Relations involving one angle.- 1.11.1.2 Sums and differences of two angles.- 1.11.1.3 Multiples of one angle.- 1.11.1.4 Functions of half angles.- 1.11.1.5 Sums and differences of hyperbolic functions.- 1.11.1.6 Products and powers of hyperbolic functions.- 1.11.2 Relations between hyperbolic and trigonometric functions.- 1.11.3 Hyperbolic functions of complex argument.- 1.11.4 The inverse hyperbolic functions.- 1.11.4.1 Fundamental relations for inverse hyperbolic functions.- 1.11.4.2 Relations between inverse hyperbolic and inverse trigonometric functions.- 1.11.5 Series expansion of hyperbolic functions.- 1.11.5.1 Series expansion of inverse hyperbolic functions.- 1.11.6 Values of hyperbolic functions.- 1.12 Algebraic equations and solutions.- 1.12.1 Linear: ax + b = 0.- 1.12.2 Quadratic: ax2 + bx + c = 0.- 1.12.3 Cubic: ax3 + bx2 + cx + d = 0.- 1.12.4 System of linear equations.- 1.12.4.1 Homogeneous system of linear equations.- 1.12.4.2 Inhomogeneous system of linear equations.- 1.13 Calculus.- 1.13.1 Derivatives.- 1.13.1.1 Derivatives of elementary functions.- 1.13.1.2 Higher derivatives of elementary functions.- 1.13.2 Elementary rules of integration.- 1.13.3 Integrals.- 1.13.3.1 Definite integrals.- 1.13.3.2 Elliptic integrals.- 1.13.3.3 Complete elliptical integrals.- 1.14 Series.- 1.14.1 Finite series.- 1.14.2 Infinite series.- 1.14.2.1 Sums of reciprocal powers.- 1.14.3 Binomial series.- 1.14.3.1 Binomial coefficients.- 1.14.4 Fourier series.- 1.14.5 Taylor series.- 1.14.6 MacLaurin series.- 1.15 Polynomials.- 1.15.1 Bernoulli's polynomials and Bernoulli's numbers.- 1.15.2 Legendre's polynomials (First kind).- 1.15.2.1 Associated Legendre's polynomials (First kind).- 1.15.3 Hermite polynomials.- 1.16 Special functions.- 1.16.1 Error function.- 1.16.1.1 Error function for complex argument.- 1.16.2 Bessel functions.- 1.16.3 Gamma and Beta function.- 1.17 Fourier transformation.- 1.17.1 Fourier cosine transforms.- 1.17.2 Fourier sine transforms.- 1.17.3 Exponential Fourier transforms.- 1.18 Vectors and vector analysis.- 1.18.1 Unit vectors defining right handed orthogonal systems.- 1.18.1.1 Rectangular coordinates (x,y,z).- 1.18.1.2 Cylindrical coordinates (r,?,z).- 1.18.1.3 Spherical coordinates (R,?,?).- 1.18.1.4 Transformation between unit vector systems.- 1.18.1.5 Derivatives of unit vectors in polar coordinates.- 1.18.1.6 Time derivatives of unit vectors in polar coordinates.- 1.18.2 Line elements.- 1.18.3 Area elements.- 1.18.4 Position vector.- 1.18.5 Vector algebra.- 1.18.5.1 Vector sum.- 1.18.5.2 Scalar or "dot" product.- 1.18.5.3 Vector or "cross" product.- 1.18.6 Derivatives of vectors.- 1.18.6.1 Time derivative of vectors.- 1.18.7 Vector differential operator $$ \vec \nabla $$ (del).- 1.18.7.1 Gradient.- 1.18.7.2 Divergence.- 1.18.7.3 Rotation or curl.- 1.18.7.4 Identities involving the vector operator $$ \vec \nabla $$.- 1.18.7.5 Laplacian operator $$ {\vec \nabla ^2} $$.- 1.18.7.6 Identities involving line and surface integrals.- 1.18.7.7 Identities involving surface and volume integrals.- 1.19 Solution of important differential equations.- 1.19.1 Harmonic oscillator.- 1.19.1.1 Oscillator with damping.- 1.19.2 Laplace equation.- 1.19.2.1 Laplace equation in rectangular coordinates.- 1.19.2.2 Laplace equation in polar coordinates.- 1.19.2.3 Example for the solution of the Laplace equation.- 1.19.3 The wave equation.- 1.19.3.1 Telegraph equation (Wave equation in a conducting medium).- 1.19.4 Heat conduction or diffusion equation (linear).- 2 Units, conversion factors and constants.- 2.1 The international system of units (SI).- 2.1.1 The basic SI units.- 2.1.2 Supplementary SI units.- 2.1.3 Derived SI units with special names.- 2.1.4 Commonly used derived SI units without special names.- 2.1.5 Prefixes for multiples and fractions of SI units.- 2.1.6 Multiples and fractions of SI units with special names but not part of the system.- 2.1.7 Other units exactly defined in terms of SI units.- 2.1.8 Units used in the international system whose values are obtained experimentally.- 2.1.9 Symbols for SI units in alphabetical order.- 2.1.10 Symbols for derived SI units in alphabetical order.- 2.2 Conversion factors listed by physical quantity.- 2.2.1 Length - macroscopic.- 2.2.1.1 Length - microscopic.- 2.2.2 Area.- 2.2.3 Volume.- 2.2.4 Mass.- 2.2.5 Speed.- 2.2.6 Force.- 2.2.7 Pressure.- 2.2.8 Power.- 2.2.9 Energy.- 2.2.9.1 Energy - atomic units.- 2.2.9.2 Energy equivalents.- 2.2.9.3 Energy equivalents in terms of the Rydberg constant.- 2.2.9.4 Energy or voltage to wavelength conversion.- 2.2.10 Temperature.- 2.2.11 Electromagnetic quantities.- 2.2.11.1 Fundamental electromagnetic relations in Gaussian and SI units.- 2.2.12 Factors for converting from half-life to decay constant.- 2.2.13 Time unit conversion.- 2.3 Conversion factors in alphabetical order.- 2.4 Atomic units.- 2.4.1 Values of frequently occurring quantities in at…