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Useful Review on the Exponential-Integral Special Function

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Useful Review on the Exponential-Integral Special Function A review on the Exponential-Integral special function and other strictly related special functions. Lectures from a seminar of Mathematical Physics by Enrico Masina† † Dipartimento di Fisica ed Astronomia (DIFA), University of Bologna Alma Mater Studiorum, and INFN, Italy [email protected] arXiv:1907.12373v1 [math.GM] 24 Jul 2019 Introduction The aim of this short series of lectures is to provide a good treatise about the Exponential Integral special function, covering also a few selected topics like its generalisation, other related special functions (Sine Integral, Cosine Integral and the Logarithmic Integral) and the theory of Asymptotic Series. Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum Mechanics, Thermodynamics, Fluid Dynamics, Viscoelasticity, et cetera. The theory of Special Functions is closely related to the theory of Lie groups and Lie alge- bras, as well as certain topics in Mathematical Physics, hence their use and a deep knowl- edge about them is as necessary as mandatory. In its first part, the present paper aims to give a particular detailed treatise over the Ex- ponential Integral function and strictly related functions: its generalisation (the so called generalised-Ei), and the Modified Exponential Integral (often named Ein), giving their Se- ries expansion and showing their plots over the real plane. The second part is entirely dedicated to a review of Asymptotic Series, starting with the Big-O and Little-o symbols (the Landau symbols), in order to reconnect to the Exponential Integral (and related functions) to study their asymptotic behaviour. The end of this part will also look at what happens when the argument of the Exp-Integral becomes imaginary whence the study of the Sine Integral (Si) and Cosine Integral (Ci) special functions. The third part will focus on another special function strictly connected to Ei: the Logarith- mic Integral providing also a brief recall of its connection with the Prime Numbers counting function π(x ). The last part will naively present a solved physical problem where the Exponential Inte- gral pops out, eventually leaving to the readers three good exercises in order to have some practice and become familiar with those important special functions. 1 Contents 1 The ExponentialIntegral Function 3 1.1 Series Expansion for Ei(z ) ...................................... 6 1.2 The Modified Exponential Integral : Ein(z ) .......................... 12 1.3 Series Expansion for Ein(z ) ..................................... 12 2 AsymptoticSeries 14 2.1 OrderNotation.................................... .......... 14 2.1.1 Asymptotically Bounded - Big O - ......................... 14 O 2.1.2 Asymptotically Smaller - Little o - o .......................... 15 2.1.3 Asymptotically Equal - Goes Like - ......................... 15 ∼ 2.1.4 Properties of asymptotic orders . ......... 16 2.1.5 UsefulExamples ................................ ....... 16 2.1.6 AsymptoticSequences ........................... ........ 17 2.1.7 AsymptoticSeries.............................. ......... 17 2.2 Asymptotic Series of Ei(z ) and (z ) ............................... 19 E1 2.3 ImaginaryArgument............................... ........... 22 2.3.1 Series Expansion of Si(x ) and Ci(x ) .......................... 25 2.3.2 Asymptotic Series for Si(x ) and Ci(x ) ......................... 26 3 The LogarithmicIntegral 27 3.1 Series of li(z ) ................................................ 28 3.2 ModifiedLogarithmicIntegral.. .. .. .. .. .. .. .. .. .. ............ 29 3.3 Asymptotic expansion of li(z ) and li1(z ) ............................ 29 3.4 OffsetLogarithmicIntegral ....................... .............. 29 3.5 Prime Numbers counter π(x ) and Li(x ) ............................ 30 4 Physical Application and Exercises 34 4.1 Application..................................... ............ 34 4.2 Exercises ....................................... ........... 36 2 Chapter 1 The Exponential Integral Function We can immediately start by giving the mathematical expression of the so called Exponen- tial Integral Function, which is defined as x e t Ei(x )= dt x W1 t ∈ Z−∞ So we can see the Exponential Integral is defined as one particular definite integral of the ratio between the exponential function and its argument. The Risch Algorithm can be performed to show that this integral is not an elementary func- tion, namely a primitive of Ei(x ) in terms of elementary functions does not exist. Such a function has a pole at t = 0, hence we shall interpret this integral as the Cauchy Principal Value: α x e t e t Ei(x )= lim dt + dt α 0 t t → Z Zα −∞ We can define Ei(x ) in a more suitable way through a parity transformation t t →− x x ¨ →− one gets + t ∞ e − (x )= Ei( x )= dt E1 − − t Zx 1The set W is defined as the Whole set, namely the set of all the natural numbers but zero: W R+ 0 = 1,2,3,4,... ≡ \{ } { } We use instead the notation R+ to indicate the set of the natural numbers plus zero: R+ = 0,1,2,3,... { } 3 When the argument takes complex values, the definition of the integral becomes ambigu- ous due to branch points at 0 and . ∞ Introducing the complex variable z = x + iy , we can use the following notation to define the Exponential Integral in the Complex plane2: + t ∞ e − (z )= dt arg(z ) <π E1 t | | Zz To clarify the graphical behaviour of the two functions, the following plot may come in handy. We can immediately write down some useful known values: Ei(0)= Ei( )= 0 Ei(+ )=+ −∞ −∞ ∞ ∞ (0)=+ (+ )= 0 ( )= E1 ∞ E1 ∞ E1 −∞ −∞ 2A brief summary about complex numbers: Given a complex number z = x + iy we can always write it in the exponential form z = z e i θ | | where y z = x 2 + y 2 θ = arg(z )= arctan | | x p 4 It’s actually simple to find the values of (x ) from Ei(x ) (and vice versa) by using the pre- E1 viously written relation: (x )= Ei( x ) E1 − − We notice that the function (x ) is a monotonically decreasing function in the range (0;+ ). E1 ∞ The function (z ) is actually nothing but the so called Incomplete Gamma Function: 3 E1 (z ) Γ (0, z ) E1 ≡ where + ∞ s 1 t Γ (s, z )= t − e − dt Zz Indeed by putting s = 0 we immediately find (z ). E1 By introducing the small Incomplete Gamma Function z s 1 t γ(s, z )= t − e − dt Z0 we are able to write down a very obvious, sometimes useful and straightforward relation between the three Gamma functions: Γ (s,0)= γ(s, z )+ Γ (s, z ) Let’s now come back to the Exponential Integral. Let’s perform a naive change of variable t zu dt = z du → Step by step we get: + t + zu + zu ∞ e − ∞ e − ∞ e − dt z du = du t → zu u Zz Z1 Z1 In this way we define the General Exponential Integral: + zu ∞ e − (z )= du n R En u n ∈ Z1 with the particular value 1 (0)= En n 1 − 3Let’s recall the definition of the Ordinary Gamma Function + ∞ s 1 t Γ (s)= Γ (s,0)= t − e − dt Z0 whilst + t ∞ e − Γ (0,0)= dt = ∞ Z0 t 5 We can show the behaviour of the first five functions (x ), namely for n = 0,...5: En 1.1 SeriesExpansion for Ei(z ) Let’s go back to the Exponential Integral we defined at the very beginning of the paper. z e t Ei(z )= dt Z t −∞ What we are going to do is to split the integral initially into three pieces: z 1 0 z e t − e t e t e t dt = dt + dt + dt t t t t Z Z Z 1 Z0 −∞ −∞ − and now with a mathematical trick we add and cut the term z dt 1 t Z− as follows: 1 0 z z z − e t e t e t dt dt dt + dt + dt + − t 1 t 0 t 1 t 1 t Z−∞ Z− Z Z− Z− 0 z Expanding the last integral into + we get 1 0 − 1 0 R R z z 0 z − e t e t e t dt dt dt dt + dt + dt + − − t 1 t 0 t 1 t 1 t 0 t Z−∞ Z− Z Z− Z− Z 6 This appears as a weird mathematical manipulation but now we can add up the similar integral, namely the second term from the left with the second to last term, and the third term from the left with the very last term at the right side, getting: 1 0 z z − e t e t 1 e t 1 dt dt + − dt + − dt + t 1 t 0 t 1 t Z−∞ Z− Z Z− Now it’s about a bit of maths. Let’s take the very first term, and let’s perform the reciprocal transformation of variable t 1/t : → 1 t 0 1/t − e t 1/t e dt → dt −−−−→ t 1 t Z−∞ Z− We did this because now we can sum this term with the second term above which leads us to 0 e 1/t + e t 1 − dt 1 t Z− and again with a parity transformation t t we write it down in a well known form (this →− is a really important and interesting integral): 1 t 1/t 1 e − e − − − dt = γ = ψ(1) t − Z0 where ψ(z ) is the logarithmic derivative of the Gamma function a.k.a. the Digamma Func- tion: d ψ(z )= lnΓ (z ) dz and γ is the famous Euler-Mascheroni constant γ 0.5772156649015328606065... ≈ Brief digression over ψ(z ), γ and Γ (z ) Having introduced the Euler-Mascheroni constant, and the Digamma Function, it’s neces- sary to show some connections between them. First of all, let’s recall the Gamma function and its derivative: + ∞ z 1 t Γ (z )= t − e − dt Z0 + ∞ d z 1 t Γ ′(z )= Γ (z )= t − e − ln(t ) dt dz Z0 We define the Digamma function as Γ ′(z ) d ψ(z )= = lnΓ (z ) Γ (z ) dz 7 which is actually generally written as ψ0 (z ) because it belongs to a general class of functions called Polygamma Functions: dm+1 ψm (z )= lnΓ (z ) dz m+1 in which we recognize the Digamma function ψ0(z ) for m = 0.
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