Natural Logarithms and Antilogarithms | 24 EXPONENTIAL AND LOGARITHMIC FUNCTIONS NATURAL LOGARITHMS AND ANTILOGARITHMS Natural logarithms and antilogarithms also called Napierian are those in which the base a e . . see page 1 . The natural logarithm of N is denoted by logeN or In N. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms . tables giving ex for values of æ see pages 226-227. For illustrations using these tables see pages 196 and 200. CHANGE OF BASE OF LOGARITHMS The relationship between logarithms of a number N to different bases a and b is given by XT loge N loga N logb a In particular logcN InN 50929 94. log10 N log10 N log N 44819 03. loge N RELATIONSHIP BETWEEN EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS eie cose i sin 9. e ie cose t These are called Euler s identities. sine cose tan e cot e sec e esc e Here i is the imaginary unit see page 21 2Ï e ie 2 e 0 _ e-ie _ . eie e- 0 i ei0 e-ie - -î ei0 e w j gie e 0 i ye 0 e- 0 2 e 0 e ie 2i g ie PERIODICITY OF EXPONENTIAL FUNCTIONS gi e 2k7r eie fc integer From this it is seen that ex has period 2iri. EXPONENTIAL AND LOGARITHMIC FUNCTIONS 25 POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL The polar form of a complex number x iy can be written in terms of exponentials see page 22 as x iy r cos 9 i sin 9 re e OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM Formulas through on page 22 are equivalent to the following. r1ei0i r2eiöa r ei9i r r2e 9z r2 rei9 p rpeip9 De Moivre s theorem fl 2k7r Jl n 4- 2ktT n LOGARITHM OF A COMPLEX NUMBER In re 9 In r i9 2fari k integer 8 HYPERBOLIC FUNCTIONS DEFINITION OF HYPERBOLIC FUNCTIONS Hyperbolic sine of X sinh X ex - e x 2 Hyperbolic cosine of X cosh X _ ex e x 2 Hyperbolic tangent of X tanh X _ ex e x ex e x Hyperbolic cotangent of X coth X Hyperbolic secant of X sech X Hyperbolic cosecant of x csch X x _ e c .
