Vector Spaces in Physics 8/16/2014 A - 1 Appendix A. Useful Mathematical Facts and Formulae 1. Complex Numbers Complex numbers in general have both a real and an imaginary part. Here "i" represents the square root of -1. If C represents a complex number, it can be written C A i B, (AA-1) with A and B real numbers. Thus,   ReImACBC (AA-2) There is another representation of a complex number, in terms of its magnitude  and phase : iCe, (AA-3) There is a very useful relation between the complex exponential representation and the real trigonometric functions, the Euler equation: cos siniei (AA-4) and the inverse relations, cos2sin2iiiieeeei (AA-5) From equation (AA-4) one can deduce some useful special values for the complex exponential: 211iiee (AA-6) And from equation (AA-5) one easily deduces   cos cossin sinAAAA   (AA-7) 2. An Integrals and Two Identities. 2ue du (AB-1)   sin sin cos sin coscos cos cos sin sinA B A B B AA B A B A B     (AB-2) 3. Power Series and the Small-Angle Approximation. It is especially convenient to expand a function of a dimensionless variable as a power series when the variable can be taken to be reasonably small. Some useful power series are given below.Vector Spaces in Physics 8/16/2014 A - 2    3524232 3 1sin ... ...3! 5!cos 1 .. ...2! 41 ...2! 3!12(11 1 ...2! 3!xnnnxxxxxxxxxexn n nnnx nx x x nx x                    (AC-1) The small-angle approximation usually amounts to keeping terms up through the first non-constant term, as below.  2121211111sincos 11111111nxxxxxxxx nx       (AC-2) 4. Mathematical Logical Symbols. Some symbols used in mathematical logic make definitions and other mathematical discussions easier. Here are some that I might use in class.  there exists  for all or for every  is contained in (as an element of a set)  such that 