Spectral Zeta Functions of Graphs and the Riemann Zeta Function in the Critical Strip
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Riemann Sums Fundamental Theorem of Calculus Indefinite
Riemann Sums Partition P = {x0, x1, . , xn} of an interval [a, b]. ck ∈ [xk−1, xk] Pn R(f, P, a, b) = k=1 f(ck)∆xk As the widths ∆xk of the subintervals approach 0, the Riemann Sums hopefully approach a limit, the integral of f from a to b, written R b a f(x) dx. Fundamental Theorem of Calculus Theorem 1 (FTC-Part I). If f is continuous on [a, b], then F (x) = R x 0 a f(t) dt is defined on [a, b] and F (x) = f(x). Theorem 2 (FTC-Part II). If f is continuous on [a, b] and F (x) = R R b b f(x) dx on [a, b], then a f(x) dx = F (x) a = F (b) − F (a). Indefinite Integrals Indefinite Integral: R f(x) dx = F (x) if and only if F 0(x) = f(x). In other words, the terms indefinite integral and antiderivative are synonymous. Every differentiation formula yields an integration formula. Substitution Rule For Indefinite Integrals: If u = g(x), then R f(g(x))g0(x) dx = R f(u) du. For Definite Integrals: R b 0 R g(b) If u = g(x), then a f(g(x))g (x) dx = g(a) f( u) du. Steps in Mechanically Applying the Substitution Rule Note: The variables do not have to be called x and u. (1) Choose a substitution u = g(x). du (2) Calculate = g0(x). dx du (3) Treat as if it were a fraction, a quotient of differentials, and dx du solve for dx, obtaining dx = . -
Tsinghua Lectures on Hypergeometric Functions (Unfinished and Comments Are Welcome)
Tsinghua Lectures on Hypergeometric Functions (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] December 8, 2015 1 Contents Preface 2 1 Linear differential equations 6 1.1 Thelocalexistenceproblem . 6 1.2 Thefundamentalgroup . 11 1.3 The monodromy representation . 15 1.4 Regular singular points . 17 1.5 ThetheoremofFuchs .. .. .. .. .. .. 23 1.6 TheRiemann–Hilbertproblem . 26 1.7 Exercises ............................. 33 2 The Euler–Gauss hypergeometric function 39 2.1 The hypergeometric function of Euler–Gauss . 39 2.2 The monodromy according to Schwarz–Klein . 43 2.3 The Euler integral revisited . 49 2.4 Exercises ............................. 52 3 The Clausen–Thomae hypergeometric function 55 3.1 The hypergeometric function of Clausen–Thomae . 55 3.2 The monodromy according to Levelt . 62 3.3 The criterion of Beukers–Heckman . 66 3.4 Intermezzo on Coxeter groups . 71 3.5 Lorentzian Hypergeometric Groups . 75 3.6 Prime Number Theorem after Tchebycheff . 87 3.7 Exercises ............................. 93 2 Preface The Euler–Gauss hypergeometric function ∞ α(α + 1) (α + k 1)β(β + 1) (β + k 1) F (α, β, γ; z) = · · · − · · · − zk γ(γ + 1) (γ + k 1)k! Xk=0 · · · − was introduced by Euler in the 18th century, and was well studied in the 19th century among others by Gauss, Riemann, Schwarz and Klein. The numbers α, β, γ are called the parameters, and z is called the variable. On the one hand, for particular values of the parameters this function appears in various problems. For example α (1 z)− = F (α, 1, 1; z) − arcsin z = 2zF (1/2, 1, 3/2; z2 ) π K(z) = F (1/2, 1/2, 1; z2 ) 2 α(α + 1) (α + n) 1 z P (α,β)(z) = · · · F ( n, α + β + n + 1; α + 1 − ) n n! − | 2 with K(z) the Jacobi elliptic integral of the first kind given by 1 dx K(z)= , Z0 (1 x2)(1 z2x2) − − p (α,β) and Pn (z) the Jacobi polynomial of degree n, normalized by α + n P (α,β)(1) = . -
The Definite Integrals of Cauchy and Riemann
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Analysis Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2017 The Definite Integrals of Cauchy and Riemann Dave Ruch [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_analysis Part of the Analysis Commons, Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Ruch, Dave, "The Definite Integrals of Cauchy and Riemann" (2017). Analysis. 11. https://digitalcommons.ursinus.edu/triumphs_analysis/11 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Analysis by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Definite Integrals of Cauchy and Riemann David Ruch∗ August 8, 2020 1 Introduction Rigorous attempts to define the definite integral began in earnest in the early 1800s. A major motivation at the time was the search for functions that could be expressed as Fourier series as follows: 1 a0 X f (x) = + (a cos (kw) + b sin (kw)) where the coefficients are: (1) 2 k k k=1 1 Z π 1 Z π 1 Z π a0 = f (t) dt; ak = f (t) cos (kt) dt ; bk = f (t) sin (kt) dt. 2π −π π −π π −π P k Expanding a function as an infinite series like this may remind you of Taylor series ckx from introductory calculus, except that the Fourier coefficients ak; bk are defined by definite integrals and sine and cosine functions are used instead of powers of x. -
Transformation Formulas for the Generalized Hypergeometric Function with Integral Parameter Differences
CORE Metadata, citation and similar papers at core.ac.uk Provided by Abertay Research Portal Transformation formulas for the generalized hypergeometric function with integral parameter differences A. R. Miller Formerly Professor of Mathematics at George Washington University, 1616 18th Street NW, No. 210, Washington, DC 20009-2525, USA [email protected] and R. B. Paris Division of Complex Systems, University of Abertay Dundee, Dundee DD1 1HG, UK [email protected] Abstract Transformation formulas of Euler and Kummer-type are derived respectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), where r pairs of numeratorial and denominatorial parameters differ by positive integers. Certain quadratic transformations for the former function, as well as a summation theorem when x = 1, are also considered. Mathematics Subject Classification: 33C15, 33C20 Keywords: Generalized hypergeometric function, Euler transformation, Kummer trans- formation, Quadratic transformations, Summation theorem, Zeros of entire functions 1. Introduction The generalized hypergeometric function pFq(x) may be defined for complex parameters and argument by the series 1 k a1; a2; : : : ; ap X (a1)k(a2)k ::: (ap)k x pFq x = : (1.1) b1; b2; : : : ; bq (b1)k(b2)k ::: (bq)k k! k=0 When q = p this series converges for jxj < 1, but when q = p − 1 convergence occurs when jxj < 1. However, when only one of the numeratorial parameters aj is a negative integer or zero, then the series always converges since it is simply a polynomial in x of degree −aj. In (1.1) the Pochhammer symbol or ascending factorial (a)k is defined by (a)0 = 1 and for k ≥ 1 by (a)k = a(a + 1) ::: (a + k − 1). -
On the Approximation of Riemann-Liouville Integral by Fractional Nabla H-Sum and Applications L Khitri-Kazi-Tani, H Dib
On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L Khitri-Kazi-Tani, H Dib To cite this version: L Khitri-Kazi-Tani, H Dib. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications. 2016. hal-01315314 HAL Id: hal-01315314 https://hal.archives-ouvertes.fr/hal-01315314 Preprint submitted on 12 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L. Khitri-Kazi-Tani∗ H. Dib y Abstract First of all, in this paper, we prove the convergence of the nabla h- sum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case. Keywords. Riemann-Liouville integral, nabla h-sum, fractional derivatives, fractional differential equation, time scales convergence MSC (2010). 26A33, 34A08, 34K28, 34N05 1 Introduction The numerous applications of fractional calculus in the mathematical modeling of physical, engineering, economic and chemical phenomena have contributed to the rapidly growing of this theory [12, 19, 21, 18], despite the discrepancy on definitions of the fractional operators leading to nonequivalent results [6]. -
And Ξ(T) Functions Associated with Riemann's Ζ(S)
Some results on the ξ(s) and Ξ(t) functions associated with Riemann’s ζ(s) function. ∗ Hisashi Kobayashi† 2016/03/04 Abstract 1 We report on some properties of the ξ(s) function and its value on the critical line, Ξ(t)= ξ 2 + it. First, we present some identities that hold for the log derivatives of a holomorphic function. We then re- examine Hadamard’s product-form representation of the ξ(s) function, and present a simple proof of the horizontal monotonicity of the modulus of ξ(s). We then show that the Ξ(t) function can be interpreted as the autocorrelation function of a weakly stationary random process, whose power spectral function S(ω) and Ξ(t) form a Fourier transform pair. We then show that ξ(s) can be formally written as the − 1 Fourier transform of S(ω) into the complex domain τ = t iλ, where s = σ + it = 2 + λ + it. We then show that the function S1(ω) studied by P´olya has g(s) as its Fourier transform, where ξ(s)= g(s)ζ(s). Finally we discuss the properties of the function g(s), including its relationships to Riemann-Siegel’s ϑ(t) function, Hardy’s Z-function, Gram’s law and the Riemann-Siegel asymptotic formula. Key words: Riemann’s ζ(s) function, ξ(s) and Ξ(t) functions, Riemann hypothesis, Monotonicity of the modulus ξ(t), Hadamard’s product formula, P´olya’s Fourier transform representation, Fourier transform to the complex domain, Riemann-Siegel’s asymptotic formula, Hardy’s Z-function. -
Computation of Hypergeometric Functions
Computation of Hypergeometric Functions by John Pearson Worcester College Dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematical Modelling and Scientific Computing University of Oxford 4 September 2009 Abstract We seek accurate, fast and reliable computations of the confluent and Gauss hyper- geometric functions 1F1(a; b; z) and 2F1(a; b; c; z) for different parameter regimes within the complex plane for the parameters a and b for 1F1 and a, b and c for 2F1, as well as different regimes for the complex variable z in both cases. In order to achieve this, we implement a number of methods and algorithms using ideas such as Taylor and asymptotic series com- putations, quadrature, numerical solution of differential equations, recurrence relations, and others. These methods are used to evaluate 1F1 for all z 2 C and 2F1 for jzj < 1. For 2F1, we also apply transformation formulae to generate approximations for all z 2 C. We discuss the results of numerical experiments carried out on the most effective methods and use these results to determine the best methods for each parameter and variable regime investigated. We find that, for both the confluent and Gauss hypergeometric functions, there is no simple answer to the problem of their computation, and different methods are optimal for different parameter regimes. Our conclusions regarding the best methods for computation of these functions involve techniques from a wide range of areas in scientific computing, which are discussed in this dissertation. We have also prepared MATLAB code that takes account of these conclusions. -
A Note on the 2F1 Hypergeometric Function
A Note on the 2F1 Hypergeometric Function Armen Bagdasaryan Institution of the Russian Academy of Sciences, V.A. Trapeznikov Institute for Control Sciences 65 Profsoyuznaya, 117997, Moscow, Russia E-mail: [email protected] Abstract ∞ α The special case of the hypergeometric function F represents the binomial series (1 + x)α = xn 2 1 n=0 n that always converges when |x| < 1. Convergence of the series at the endpoints, x = ±1, dependsP on the values of α and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series for |x| < 1 and obtain new result on its convergence at point x = −1 for every integer α = 0, that is we prove it for the function 2F1(α, β; β; x). The proof is within a new theoretical setting based on a new method for reorganizing the integers and on the original regular method for summation of divergent series. Keywords: Hypergeometric function, Binomial series, Convergence radius, Convergence at the endpoints 1. Introduction Almost all of the elementary functions of mathematics are either hypergeometric or ratios of hypergeometric functions. Moreover, many of the non-elementary functions that arise in mathematics and physics also have representations as hypergeometric series, that is as special cases of a series, generalized hypergeometric function with properly chosen values of parameters ∞ (a ) ...(a ) F (a , ..., a ; b , ..., b ; x)= 1 n p n xn p q 1 p 1 q (b ) ...(b ) n! n=0 1 n q n X Γ(w+n) where (w)n ≡ Γ(w) = w(w + 1)(w + 2)...(w + n − 1), (w)0 = 1 is a Pochhammer symbol and n!=1 · 2 · .. -
Lauricella's Hypergeometric Function
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 452-470 (1963) Lauricella’s Hypergeometric Function FD* B. C. CARLSON Institute for Atomic Research and D@artment of Physics, Iowa State University, Ames, Iowa Submitted by Richard Bellman I. INTRODUCTION In 1880 Appell defined four hypergeometric series in two variables, which were generalized to 71variables in a straightforward way by Lauricella in 1893 [I, 21. One of Lauricella’s series, which includes Appell’s function F, as the case 71= 2 (and, of course, Gauss’s hypergeometric function as the case n = l), is where (a, m) = r(u + m)/r(a). The FD f unction has special importance for applied mathematics and mathematical physics because elliptic integrals are hypergeometric functions of type FD [3]. In Section II of this paper we shall define a hypergeometric function R(u; b,, . ..) b,; zi, “., z,) which is the same as FD except for small but important modifications. Since R is homogeneous in the variables zi, ..., z,, it depends in a nontrivial way on only 71- 1 ratios of these variables; indeed, every R function with n variables is expressible in terms of an FD function with 71- 1 variables, and conversely. Although R and FD are therefore equivalent, R turns out to be more convenient for both theory and applica- tion. The choice of R was initially suggested by the observation that the Euler transformations of FD would be greatly simplified by introducing homo- geneous variables. -
Using the TI-89 to Find Riemann Sums
Using the TI-89 to find Riemann sums Z b If function f is continuous on interval [a, b], f(x) dx exists and can be approximated a using either of the Riemann sums n n X X f(xi)∆x or f(xi−1)∆x i=1 i=1 (b−a) where ∆x = n and n is the number of subintervals chosen. The first of these Riemann sums evaluates function f at the right endpoint of each subinterval; the second evaluates at the left endpoint of each subinterval. n X P To evaluate f(xi) using the TI-89, go to F3 Calc and select 4: ( sum i=1 The command line should then be completed in the following form: P (f(xi), i, 1, n) The actual Riemann sum is then determined by multiplying this ans by ∆x (or incorpo- rating this multiplication into the same command line.) [Warning: Be sure to set the MODE as APPROXIMATE in this case. Otherwise, the calculator will spend an inordinate amount of time attempting to express each term of the summation in exact symbolic form.] Z 4 √ Example: To approximate 1 + x3 dx using Riemann sums with n = 100 subinter- 2 b−a 2 i vals, note first that ∆x = n = 100 = .02. Also xi = 2 + i∆x = 2 + 50 . So the right endpoint approximation will be found by entering √ P( (2 + (1 + i/50)∧3), i, 1, 100) × .02 which gives 10.7421. This is an overestimate, since the function is increasing on the given interval. To find the left endpoint approximation, replace i by (i − 1) in the square root part of the command. -
Numerical Evaluation of the Gauss Hypergeometric Function with the Hypergeo Package
Numerical evaluation of the Gauss hypergeometric function with the hypergeo package Robin K. S. Hankin Auckland University of Technology Abstract This paper introduces the hypergeo package of R routines, for numerical calculation of hypergeometric functions. The package is focussed on efficient and accurate evaluation of the hypergeometric function over the whole of the complex plane within the constraints of fixed-precision arithmetic. The hypergeometric series is convergent only within the unit circle, so analytic continuation must be used to define the function outside the unit circle. This short document outlines the numerical and conceptual methods used in the package; and justifies the package philosophy, which is to maintain transparent and verifiable links between the software and AMS-55. The package is demonstrated in the context of game theory. Keywords: Hypergeometric functions, numerical evaluation, complex plane, R, residue theo- rem. 1. Introduction k The geometric series k∞=0 tk with tk = z may be characterized by its first term and the con- k 1 stant ratio of successiveP terms tk+1/tk = z, giving the familiar identity k∞=0 z = (1 − z)− . Observe that while the series has unit radius of convergence, the rightP hand side is defined over the whole complex plane except for z = 1 where it has a pole. Series of this type may be generalized to a hypergeometric series in which the ratio of successive terms is a rational function of k: t +1 P (k) k = tk Q(k) where P (k) and Q(k) are polynomials. If both numerator and denominator have been com- pletely factored we would write t +1 (k + a1)(k + a2) ··· (k + ap) k = z tk (k + b1)(k + b2) ··· (k + bq)(k + 1) (the final term in the denominator is due to historical reasons), and if we require t0 = 1 then we write ∞ k a1,a2,...,ap tkz = aFb ; z (1) b1,b2,...,bq Xk=0 2 Numerical evaluation of the Gauss hypergeometric function with the hypergeo package − z when defined. -
Fractional Calculus
Generalised Fractional Calculus Penelope Drastik Supervised by Marianito Rodrigo September 2018 Outline 1. Integration 2. Fractional Calculus 3. My Research Riemann sum: P tagged partition, f :[a; b] ! R n X S(f ; P) = f (ti )(xi − xi−1) i=1 The Riemann sum Partition of [a; b]: Non-overlapping intervals which cover [a; b] a = x0 < x1 < ··· < xi−1 < xi < ··· xn = b Tagged partition of [a; b]: Intervals in partition paired with tag points n P = f([xi−1; xi ]; ti )gi=1 The Riemann sum Partition of [a; b]: Non-overlapping intervals which cover [a; b] a = x0 < x1 < ··· < xi−1 < xi < ··· xn = b Tagged partition of [a; b]: Intervals in partition paired with tag points n P = f([xi−1; xi ]; ti )gi=1 Riemann sum: P tagged partition, f :[a; b] ! R n X S(f ; P) = f (ti )(xi − xi−1) i=1 The Riemann integral R b a f (x)dx = A means that: For all " > 0, there exists δ" > 0 such that if a tagged partition P satisfies 0 < xi − xi−1 < δ" for i = 1;:::; n then jS(f ; P) − Aj < " The Generalised Riemann integral R b a f (x)dx = A means that: For all " > 0, there exists a strictly positive function δ" :[a; b] ! R+ such that if a tagged partition P satisfies ti 2 [xi−1; xi ] ⊆ [ti − δ(ti ); ti + δ(ti )] for i = 1;:::; n then jS(f ; P) − Aj < " Riemann-Stieltjes integral: R b a f (x)dα(x) = A means that: For all " > 0, there exists δ" > 0 such that if a tagged partition P satisfies 0 < xi − xi−1 < δ" for i = 1;:::; n then jS(f ; P; α) − Aj < " The Riemann-Stieltjes integral Riemann-Stieltjes sum of f : Integrator α : I ! R, partition P n X S(f ; P; α) = f (ti )[α(xi