By Helmuth Euler

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F (x + 2l) = f (x), and that the so-called Fourier coeﬃcients ak , bk are equal to a0 = 1 2l f (x) dx, ak = 1 l f (x) cos kπx dx, l bk = 1 l f (x) sin Symmetric functions f even function, i. e. f (−x) = f (x) =⇒ bk = 0 for k = 1, 2, ... f odd function, i. e. f (−x) = −f (x) =⇒ ak = 0 for k = 0, 1, 2, ... kπx dx. l 30 Sequences and series Table of some Fourier series The functions are deﬁned on an interval of the length 2π and continued with the period 2π. x for −π < x < π 0 for x = π y = π sin x sin 2x sin 3x − + ± ...

0 π 2π 3π Mathematics of Finance Simple interest Notations p – interest rate per conversion period (in percent) t – time in units that correspond to the rate, moment K0 – opening capital, original principal, present value Kt – amount, capital at the moment t Zt – simple interest earned in the term t i – interest rate: i = T – number of days p 100 • The most frequently period considered is the year, but a half-year, quarter, month etc.

A) Find the least common denominator and b) multiply by the least common denominator. c) The substitution of x = a, x = b, . . yields Aα , Bβ , . . d) The comparison of coeﬃcients leads to linear equations for the remaining unknown coeﬃcients. Exponential functions 51 Exponential functions a ∈ IR, a > 0 y = ax a x – – – exponential function, base exponent Special case a = e: y = ex = exp(x) – exponential function to the base e Domain: Df = IR Range: Wf = IR+ = {y | y > 0} • The inverse function of the exponential function y = ax is the logarithmic function y = loga x ( p.