Here are some evaluations for these two functions. Definition 5.8.1Let f∈H1(a,b), b>a, α∈[0,1], then the definition of the Atangana–Baleanu derivative in Caputo sense is given as:(5.134)DtαABC(f(t))=B(α)1−α∫btdf(y)dyEα[−α(t−y)α1−α]dy. By observing that, it can be verified that (1 + x)f′(x) = αf(x). Using these and substituting y = −x in (iii) now yields (cos x)2 + (sin x)2 = 1, The tricky step is to show that sin and cos are periodic. Solution Using the quotient rule, the chain rule and Theorem 15.2.5 (1). The restriction of k to being an integer in Theorem 12.9 is not essential − the result remains true if k is a positive real number. ), (ii) By using the Cauchy Product Theorem, we show that if x, y ∈ ℝ sin x cos y=∑∞n=0cn where. Hence the exponential function has no real zero. Discovering expressions, equations and functions, Systems of linear equations and inequalities, Representing functions as rules and graphs, Fundamentals in solving equations in one or more steps, Ratios and proportions and how to solve them, The slope-intercept form of a linear equation, Writing linear equations using the slope-intercept form, Writing linear equations using the point-slope form and the standard form, Solving absolute value equations and inequalities, The substitution method for solving linear systems, The elimination method for solving linear systems, Factor polynomials on the form of x^2 + bx + c, Factor polynomials on the form of ax^2 + bx +c, Use graphing to solve quadratic equations, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 Internationell-licens. (6.26) applied for p = 1 it can be shown that the polar n-dimensional cosexponential functions have the property that, for even n, and in an odd number of dimensions, with the aid of Eq. (3.523), For x = y the relations (3.534)–(3.537) take the form, For x = −y the relations (3.534)–(3.537) and (3.529) yield. This oblique approach is useful elsewhere. for k = 0, 1, …, n − 1, where Rea+ib)=a, with a and b real numbers. Before we show what else can be done with our techniques, let us prove the General Binomial Theorem. We know that x ∊ ℝ ⇒ ex ∊ (0, ∞). Then the series ∑ cnxn is absolutely convergent and. Then there exist N1, N2 and N3 such that, Let N = max(N1, N2, N3) so that all three inequalities in (1) hold if n ≥ N. Recalling that if n ≥ 2 N, all the terms (ai xi)(bj xj) which occur in sN tN occur in un (see Fig. Notice that this graph violates all the properties we listed above. Example 12.2 We can now define more of the standard functions of mathematics, in this case the principal trigonometric functions sine and cosine. We define the two functions sin and cos by, By using the ratio test we see that both series converge for all x ∈ ℝ and Theorem 12.5 shows that, (Notice that a little care is needed in that the coefficient of x0 in sin x is zero, so ddxsinx=∑n=0∞−1n2n+1x2n2n+1!. □. To find the sum of ∑ ckxk we need more effort. In earlier chapters we talked about the square root as well. Note that this implies that \({b^x} \ne 0\). This allows us to produce the required properties of the exponential function: The exponential function is continuous, possesses derivatives of all orders and ddxexpx=expx. The number e is about 2.71828…; the series converges so rapidly that it is easy to evaluate it to a high accuracy. It can be checked that the derivatives of the polar cosexponential functions are related by, George B. Arfken, ... Frank E. Harris, in Mathematical Methods for Physicists (Seventh Edition), 2013, Taylor series are often used in situations where the reference point, a, is assigned the value zero. Now, let’s take a look at a couple of graphs. Sometimes we’ll see this kind of exponential function and so it’s important to be able to go between these two forms.