Vector Spaces in Physics 8/12/2010 A - 1 Appendix A. Useful Mathematical Facts and Formulae A. 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, ()( )ReImA CB C==, (AA-2) There is another representation of a complex number, in terms of its magnitude ρ and phase φ: iC eφρ=, (AA-3) There is a very useful relation between the complex exponential representation and the real trigonometric functions, the Euler equation: cos sinie iθθ θ= + (AA-4) and the inverse relations, cos2sin2i ii ie ee eiθ θθ θθθ−−+=−= (AA-5) From equation (AA-4) one can deduce some useful special values for the complex exponential: 211iieeππ= −= (AA-6) B. Some Integrals and Identities. 2ue duπ∞−∞−=∫ (AB-1) ()( )sin sin cos sin coscos cos cos sin sinA B A B B AA B A B A B+ = ++ = − (AB-2) ()( )cos cossin sinA AA A− =− = − (AB-3) C. The Small-Angle Approximation. This is really about functions of any dimensionless variable which is small compared to unity; the trigonometric functions are a well known special case, hence the name.Vector Spaces in Physics 8/12/2010 A - 2 ( )2121211111sincos 11111 11 1nxxxxxx xx nxθθθ θθ θ→→ −→ −++ → ++ → +≪≪≪≪≪ (AC-2) D. 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 ⇒