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Elementary Functions Received: 03-01-2018 Accepted: 04-02-2018 Sweeti Devi

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Elementary Functions Received: 03-01-2018 Accepted: 04-02-2018 Sweeti Devi International Journal of Statistics and Applied Mathematics 2018; 3(2): 07-09 ISSN: 2456-1452 Maths 2018; 3(2): 07-09 © 2018 Stats & Maths Elementary functions www.mathsjournal.com Received: 03-01-2018 Accepted: 04-02-2018 Sweeti Devi Sweeti Devi Assistant Professor, Abstract Bhagat Phool Singh Mahila Elementary function is an introduction to the college mathematics. It aims to provide a working Vishwavidyalaya, Khanpur knowledge of basic functions (exponential, logarithmic and trignometric) fast and efficient Kalan, Haryana, India implementation of elementary function such as sin0, cos0, log0 are of ample importance in a large class of applications. The state of the art method for function evaluation involve either expensive calculations such as multiplication, large number of iteration or large looks up tables. Our proposed method evaluates trignometric, hyperbolic, exponential, logarithmic. Keywords: Power series, logarithmic function, exponential function, trigonometric function Introduction 푥 푥 푥 Elementary functions are 푒 , 푙표푛 , 푎 , sinx, cosx. We all familiar with these functions, but this acquaintances is based on a treatment which was essentially based on intuitive less rigorous geometrical considerations. We shall base the study of these functions on the set of real numbers as a complete ordered field, the nation of limit and convergence of series. As the definitions of function will be based on power series, we start with a brief study of power series. Power Series 2 ℎ 8 ℎ The series of the form 푎표 + 푎 푥 + 푎2 푥 +....... + 푎푛 푥 + …..... = ∑푛=0 푎푛 푥 Are called power series in x and the numbers are their co-efficient. Some simple examples of the power series are. 1. The geometric series is convergent for ∑푥 ℎ Divergent for /x/< /, divergent for /x/≥ I 푥ℎ 푥2ℎ 2. ∑ Is convergent for every real x; likewise the series ∑ (−1)ℎ 푛! 2ℎ! ℎ ℎ 3. ∑ ℎ 푥 Converges for x = o, but diverges for x ≠ o For x = o, obviously every poser ℎ series ∑ 푎푛 푥 Is convergent, whatever the value of coefficient 푎푛 for a power series ℎ ∑ 푎푛 푥 , which does not merely converge everywhere or nowhere, a definite positive number R exists such that the series converges for every /x/x LR, but diverges for every/x/7R. The number R is called Radius of Convergence and the interval]-R, R [the interval of convergence. ℎ For the power series ∑ 푎푛 푥 Lim sup series ……………….. By couch’s root test, it follows if 푛 Lim sup √19푛푙 = and R = 1. R = 0, = the series is nowhere convergent. 2. R = ∞, the series is everywhere convergent. Correspondence 3. O<R<∞, the series converges absolutely. Sweeti Devi For /x/< R, and diverges for /x/< R Assistant Professor, i.e. R is radius of convergence. Bhagat Phool Singh Mahila Vishwavidyalaya, Khanpur Kalan, Haryana, India ~7~ International Journal of Statistics and Applied Mathematics Logarithmic Functions Since L is differentiable and increasing throughout its domain It is convenient at this stage to introduce two of the most I there is an increasing differentiable inverse function 퐿1 : R important functions is mathematics. It a huge range of → (표1∞), which for the moment we shall denote by E. Thus applications, form the discharge of a capacitor to the E (L(x)) = x (x € (표1∞)) population growth of acapacitor to the population growth of L (E (x)) = x (x € R) bacteria, the exponential function plays a crucial role and “log – log” graphs are a crucial part of methodology over a wide Hence 퐸 (x) = (퐿−1 ) (x) = 1 = 퐿−1(푥) = E (x) 퐿 (퐿(푥)) area of experimental science. But first let us remind of what we mean by a logarithm. If 푥 The graph of E is given by 푎 = b Theorem – ⇒ x = 푙표푎b, the logarithm of a to a base a. In words, 푙표푎b For all 푋퐼 Y in R, is the power to which a must be raised to obtain b. there are difficulties about this definition. 푥 E(x+y) = E (x) E(y) For example we know that 푎 means only if x is rational and a related difficulty it is not clear that 푙표 b is defined for 1 푎 E(-x) = every b. 퐸(푥) However this difficulty will vanish very soon. We begin with the simple observation that the formula. Relationship between exponential function and logarithm function ℎ+1 ℎ 푥 We can see the relationship between exponential function f(x) ∫ 푥 dx = is not valid when n= -1 on the other hand the 푥 푛+1 = 푒 and the logarithm function f(x) - 퐼푛푥 by looking their 1 function x → is . graphs. 푥 We can see that straight away that the logarithm function is a Continuous in any interval [a, b] not containing ‘o’ reflection of the exponential function in the line represented Let us define a new function L by by f (x) = x. In other words, the axes have been swapped; x become f(x) and f(x) becomes x. 푥 푑푡 Key Point: The exponential function f(x) = 푒푥 is the inverse L (x) = ∫ i (x 70) 푡 of pgarithm f (x) = lgx Certain properties of L are immediate. By the fundamental Remark theorem, L is differential and Before the advent of electronic, the students carried four – figure “log table” wherever they “log tables” wherever they 퐿 (x) = went. To calculate (say) 325.7 x 48.43 one looked up the two 푥 logarithms [approximately 2.5128 and 1.6851 respectively), Thus L is increasing function for all x in (표 ∞). added the two logarithms, obtaining 4.1979 and then found 44979 Notice also that L(i) = 0. 10 by looking up a table of “antilogarithms”. The answer The crucial property of the function L is given in the is not quite accurate but for most practical purposes an error following theorem. of 0.02 %. is not significant. Properties of exponential functions in terms of logarithms - The logarithm function plucks form an expression. For this Theorem: - For all 푥 y in (표∞). L (xy) = L(x) + L(y) reason, the properties of exponent translate into proportions of logarithm. L ( ) = L (x) 푥 4. For example, we know that when we multiply two terms with a common base, we add exponents. Exponential Function (푏푥) (푏푦) = 푏푥+푦 The power series 1+푥 푥2 푥3 푥푛 + + + …………. + +……………… 1 1. Product property of Logarithms 1! 2! 3! 푛! Is everywhere convergent for real x. 푙표푏(MN) = 푙표푏푁 = 푙표푏푁 2. Quotient property of logarithms 푀 푙표 ( ) = 푙표 M - 푙표 푁 Definition 푏 푁 푏 푏 The function represented by the power series (1) is called 3. Exponent property of logarithms 푐 Exponential function, denoted provisionally by E(x), thus. 푙표푏(푀 ) = C 푙표푏M 1+푥 푥2 푥3 푥푛 E(x) = + + + …………. + +……………… 2 Each of these properties is merely a restatement of a property 1! 2! 3! 푛! of exponents. 1+1 1 1 1 E(1) = + + + …………. + +……………… 3 1! 2! 3! 푛! Trigonomeiric Functions Trigonometric functions are mathematical relationships The series on the right hand side of (3) converges to a number between the angles and sides of a right triangle. The three which lies between 2 and 3. primary trigonometric functions are sinc, cosine, and tangent. This number is denoted by ‘e’, the exponential base and is the Trigonometric functions can help you. same number as represented by. All you have to do is stand in your backyard, measure the 1 Lim ( 1 + )푛 distance between yourself and the tree, and use a protractor to 푛 gauge the angel of your line of sight to the top of tree. Using ~8~ International Journal of Statistics and Applied Mathematics only these two measures. You can calculate the height of the Computation of values of the trigonometric functions free and know whether your house is safe. The values of sinx and cosx can be obtained from their Maclaurin series expansions. The expansion for sinx is Calculating the trigonometric functions 푥3 푥5 푥7 Let’s look at each of the trigonometric function and see how it Sinx = x - + - + ……. is calculated. 3! 5! 7! To calculate the sine of an angel in a right triangle you always divide the length of the side. Derivation Opposite the angle by the length of hypotenuse of angle since The series can be shown to converge for all values of x. It can is usually abbreviated as sin in mathematical statements. be used to compute the values of sinx for any value of x. The maclaurin series expansion for cosx is 푥2 푥4 −푥6 Cosx = 1 - + + ……. 2! 4! 6! Once we have computed either sinx or cosx we can compute the other through the identity 푠푛2 x + 푐표푠2x = 1. The other trigonometric functions can then be computed through relations such as. 푠푛푥 1 1 tanx = , cotx = , secx = 표푝푝표푠푡푒 푂 푐표푠푥 푡푎푛푥 푐표푠푥 푆푛 퐵 = = ℎ푦푝표푡푒푛푢푠푒 퐻 1 Cosecx = 푠푛푥 Cosing Typically cosine is abbreviated as cos, is calculated by Elementary functions when arguments is a complex number. dividing the length of the side adjacent to the angle by the We will now treat the manner in which meaning have been length of the hypotenuse of triangle. assigned to the various elementary functions when the arguments are complex numbers. The Exponential Functions 퐐퐳, 퐞퐳 We define meaning of 푎푧 when z is a complex number as follows. Definition 풆풛 푥 The maclavrin series expansion of 푒 when x is a real number is 표푝푝표푠푡푒 퐴 푐표푠 퐵 = = ℎ푦푝표푡푒푛푢푠푒 퐻 푥2 푥3 푒푥 = 1 + x + + + … 2 ! 3 ! Tangent To calculate the tangent of an angle, divide the length of the When take this maclaurin series expansion of 푒푥, x is real, as side opposite the angle by the side adjacent to the angle.
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