Gaussian Integral by Taylor Series and Applications
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Derivation: the Volume of a D-Dimensional Hypershell
Derivation: the volume of a D‐dimensional hypershell Phys 427 Yann Chemla Sept. 1, 2011 In class we worked out the multiplicity of an ideal gas by summing over volume elements in k‐space (each of which contains one quantum state) that have energy E. We showed that the surface of states that have energy E is a hypershell of radius k in D‐dimensions where . Integrating in D‐dimensions in Cartesian coordinates is easy, but how do you integrate over a hypershell in D‐ dimensions? We used dimensional arguments in class to relate a volume of k‐space in Cartesian vs. hyperspherical coordinates: Ω where Ω is the solid angle subtended by the hypershell. If we integrate over all D‐dimensional space in Cartesian vs. hyperspherical coordinates, we should get the same answer: where we integrated over the solid angle to obtain the factor gD. We would like to determine gD. The above formula is not very useful in that respect because both sides are infinite. What we want is a function that can be integrated over all of D‐dimensional space in both Cartesian and hyperspherical coordinates and give a finite answer. A Gaussian function over k1 ... kD will do the trick. We can write: where we wrote the Gaussian in terms of k on the right side of the equation. Let’s look at the left side of the equation first. The integral can be written in terms of the product of D identical 1‐dimensional integrals: / where we evaluated the Gaussian integral in the last step (look up Kittel & Kroemer Appendix A for help on evaluating Gaussian integrals). -
The Amazing Poisson Calculation: a Method Or a Trick?
The amazing Poisson calculation: a method or a trick? Denis Bell April 27, 2017 The Gaussian integral and Poisson's calculation The Gaussian function e−x2 plays a central role in probability and statistics. For this reason, it is important to know the value of the integral 1 Z 2 I = e−x dx: 0 The function e−x2 does not have an elementary antiderivative, so I cannot be evaluated by the FTC. There is the following argument, attributed to Poisson, for calculating I . Consider 1 1 Z 2 Z 2 I 2 = e−x dx · e−y dy: 0 0 Interpret this product as a double integral in the plane and transform to polar coordinates x = r cos θ; y = r sin θ to get 1 1 Z Z 2 2 I 2 = e−(x +y )dxdy 0 0 π=2 1 Z Z 2 = e−r r drdθ 0 0 1 π Z d 2 π = − e−r dr = : 4 0 dr 4 p π Thus I = 2 . Can this argument be used to evaluate other seemingly intractable improper integrals? Consider the integral Z 1 J = f (x)dx: 0 Proceeding as above, we obtain Z 1 Z 1 J2 = f (x)f (y) dxdy: 0 0 In order to take the argument further, there will need to exist functions g and h such that f (x)f (y) = g(x2 + y 2)h(y=x): Transformation to polar coordinates and the substitution u = r 2 will then yield 1 Z 1 Z π=2 J2 = g(u)du h(tan θ)dθ: (1) 2 0 0 Which functions f satisfy f (x)f (y) = g(x2 + y 2)h(y=x) (2) Theorem Suppose f : (0; 1) 7! R satisfies (2) and assume f is non-zero on a set of positive Lebesgue measure, and the discontinuity set of f is not dense in (0; 1). -
Contents 1 Gaussian Integrals ...1 1.1
Contents 1 Gaussian integrals . 1 1.1 Generating function . 1 1.2 Gaussian expectation values. Wick's theorem . 2 1.3 Perturbed gaussian measure. Connected contributions . 6 1.4 Expectation values. Generating function. Cumulants . 9 1.5 Steepest descent method . 12 1.6 Steepest descent method: Several variables, generating functions . 18 1.7 Gaussian integrals: Complex matrices . 20 Exercises . 23 2 Path integrals in quantum mechanics . 27 2.1 Local markovian processes . 28 2.2 Solution of the evolution equation for short times . 31 2.3 Path integral representation . 34 2.4 Explicit calculation: gaussian path integrals . 38 2.5 Correlation functions: generating functional . 41 2.6 General gaussian path integral and correlation functions . 44 2.7 Harmonic oscillator: the partition function . 48 2.8 Perturbed harmonic oscillator . 52 2.9 Perturbative expansion in powers of ~ . 54 2.10 Semi-classical expansion . 55 Exercises . 59 3 Partition function and spectrum . 63 3.1 Perturbative calculation . 63 3.2 Semi-classical or WKB expansion . 66 3.3 Path integral and variational principle . 73 3.4 O(N) symmetric quartic potential for N . 75 ! 1 3.5 Operator determinants . 84 3.6 Hamiltonian: structure of the ground state . 85 Exercises . 87 4 Classical and quantum statistical physics . 91 4.1 Classical partition function. Transfer matrix . 91 4.2 Correlation functions . 94 xii Contents 4.3 Classical model at low temperature: an example . 97 4.4 Continuum limit and path integral . 98 4.5 The two-point function: perturbative expansion, spectral representation 102 4.6 Operator formalism. Time-ordered products . 105 Exercises . 107 5 Path integrals and quantization . -
On Some Applicable Approximations of Gaussian Type Integrals
Journal of Mathematical Modeling Vol. 7, No. 2, 2019, pp. 221-229 JMM On some applicable approximations of Gaussian type integrals Christophe Chesneauy∗ and Fabien Navarroz yLMNO, University of Caen, Caen, France email: [email protected] zCREST, ENSAI, Rennes, France Emails: [email protected], [email protected] Abstract. In this paper, we introduce new applicable approximations for Gaussian type integrals. A key ingredient is the approximation of the func- tion e−x2 by the sum of three simple polynomial-exponential functions. Five special Gaussian type integrals are then considered as applications. Approximation of the so-called Voigt error function is investigated. Keywords: Exponential approximation, Gauss integral type function, Voigt error function. AMS Subject Classification: 26A09, 33E20, 41A30. 1 Motivation Gaussian type integrals play a central role in various branches of mathemat- ics (probability theory, statistics, theory of errors . ) and physics (heat and mass transfer, atmospheric science . ). The most famous example of this class of integrals is the Gauss error function defined by y 2 Z 2 erf(y) = p e−x dx: π 0 As for the erf(y), plethora of useful Gaussian type integral have no an- alytical expression. For this reason, a lot of approximations have been ∗Corresponding author. Received: 26 March 2019 / Revised: 28 May 2019 / Accepted: 28 May 2019. DOI: 10.22124/jmm.2019.12897.1250 c 2019 University of Guilan http://jmm.guilan.ac.ir 222 C. Chesneau, F. Navarro developed, more or less complicated, with more or less precision (for the erf(y) function, see [6] and the references therein). In this paper, we aim to provide acceptable and applicable approxi- mations for possible sophisticated Gaussian type integrals. -
Chapter 9 Integration on Rn
Integration on Rn 1 Chapter 9 Integration on Rn This chapter represents quite a shift in our thinking. It will appear for a while that we have completely strayed from calculus. In fact, we will not be able to \calculate" anything at all until we get the fundamental theorem of calculus in Section E and the Fubini theorem in Section F. At that time we'll have at our disposal the tremendous tools of di®erentiation and integration and we shall be able to perform prodigious feats. In fact, the rest of the book has this interplay of \di®erential calculus" and \integral calculus" as the underlying theme. The climax will come in Chapter 12 when it all comes together with our tremendous knowledge of manifolds. Exciting things are ahead, and it won't take long! In the preceding chapter we have gained some familiarity with the concept of n-dimensional volume. All our examples, however, were restricted to the simple shapes of n-dimensional parallelograms. We did not discuss volume in anything like a systematic way; instead we chose the elementary \de¯nition" of the volume of a parallelogram as base times altitude. What we need to do now is provide a systematic way to think about volume, so that our \de¯nition" of Chapter 8 actually becomes a theorem. We shall in fact accomplish much more than a good discussion of volume. We shall de¯ne the concept of integration on Rn. Our goal is to analyze a function f D ¡! R; where D is a bounded subset of Rn and f is bounded. -
5 Complex-Variable Theory
5 Complex-Variable Theory 5.1 Analytic functions A complex-valued function f(z) of a complex variable z is di↵erentiable at z with derivative f 0(z)ifthelimit f(z0) f(z) f 0(z)= lim − (5.1) z z z z 0! 0 − exists as z0 approaches z from any direction in the complex plane. The limit must exist no matter how or from what direction z0 approaches z. If the function f(z) is di↵erentiable in a small disk around a point z0, then f(z) is said to be analytic (or equivalently holomorphic) at z0 (and at all points inside the disk). Example 5.1 (Polynomials). If f(z)=zn for some integer n, then for tiny dz and z = z + dz,thedi↵erencef(z ) f(z)is 0 0 − n n n 1 f(z0) f(z)=(z + dz) z nz − dz (5.2) − − ⇡ and so the limit n 1 f(z0) f(z) nz − dz n 1 lim − =lim = nz − (5.3) z z z z dz 0 dz 0! 0 − ! n 1 exists and is nz − independently of how z0 approaches z. Thus the function zn is analytic at z for all z with derivative n dz n 1 = nz − . (5.4) dz 186 Complex-Variable Theory A function that is analytic everywhere is entire. All polynomials N n P (z)= cn z (5.5) n=0 X are entire. Example 5.2 (A function that’s not analytic). To see what can go wrong when a function is not analytic, consider the function f(x, y)=x2 +y2 = zz¯ for z = x + iy. -
Gaussian Integrals Depending by a Quantum Parameter in Finite
Gaussian integrals depending by a quantum parameter in finite dimension Simone Camosso∗ Abstract A common theme in mathematics is the evaluation of Gauss integrals. This, coupled with the fact that they are used in different branches of science, makes the topic always actual and interesting. In these notes we shall an- alyze a particular class of Gaussian integrals that depends by the quantum parameter ~. Starting from classical results, we will present an overview on methods, examples and analogies regarding the practice of solving quantum Gaussian integrals. Keywords. Gaussian integral, quantization, special functions, arithmetic– geometric mean, Grassmann number, Boys functions. AMS Subject Classification. 33B15, 00A05, 97I50, 81S10, 97I80, 92E10. Contents 1 Introduction 2 arXiv:2107.06874v1 [math.GM] 14 Jul 2021 2 The Gaussian integral in 1–dimension 3 2.1 WaystowritetheGaussianintegral. 7 2.2 GaussianintegralorGausstheorem? . 11 3 The Gaussian integral in n–dimensions 12 4 Gaussian integrals and homogeneous functions 14 ∗e-mail: [email protected], Saluzzo (CN) 1 5 Gaussian matrix integrals 15 6 The symplectic structure 17 7 The quantum postulates 18 8 Gaussian integrals depending by a quantum parameter 19 9 Quantum integrals 23 10 On a Gaussian integral in one dimension and the Lemniscate 24 11 The Berezin–Gaussian integral: a generalization of the Gaus- sian integral 27 11.1 On the Laplace method and a “mixed Gaussian Integral” . 29 12 Quantum Gaussian integrals with a perturbative term 30 13 Gaussian integrals and the Boys integrals 31 Bibliography 34 1 Introduction Let ~ be a quantum parameter, an important practice in quantum mechanics, is the evaluation of Gaussian integrals. -
Differentiating Under the Integral, an Alternative to Integration by Parts
Differentiating under the integral, an alternative to integration by parts Calvin W. Johnson May 9, 2020 1 The basic idea When we learn definite integrals, one of the tools is integration by parts. Here I introduce an alternative which can be applied to many of the integrals encoun- tered in physics. Called ”differentiating under the integral sign," this approach can be problematic in terms of formal integration theory, which is why math departments do not teach it, but for most smooth integrands it works just fine. It was a favorite technique of Richard Feynman. With this tool in hand, you won't have to use integrating by parts at all in this course. Differentiating under the integral sign is best explained through an example: Z 1 I = dx x exp(−ax): (1) 0 This is the kind of definite integral for which one often appeals to integration by parts. However, let's look at it carefully. We all (I hope) know Z 1 1 dx exp(−ax) = (2) 0 a or at least you know you can compute it easily enough. But notice: @ x exp(−ax) = − exp(−ax) (3) @a so that Z 1 Z 1 @ I = dx x exp(−ax) = dx − exp(−ax) (4) 0 0 @a Now because the limits of integration do not include a, we switch the order of the integral sign and the derivative. (Because both are limit processes, this is the part that is technically iffy, from a rigorous point of view, and mathematicians are correct to view it suspiciously. It does, however, yield correct results.) @ Z 1 @ 1 1 I = − dx exp(−ax) = − = 2 : (5) @a 0 @a a a 1 Huh. -
Gaussian Integrals
Gaussian integrals Calvin W. Johnson September 9, 2014 1 The basic Gaussian and its normalization The Gaussian function or the normal distribution, exp −αx2 ; (1) is a widely used function in physics and mathematical physics, including in quantum mechanics. It is therefore useful to know how to integrate it. In this and in future notes I will discuss the basic integrals you should memorize and how to derive other related integrals. An example Gaussian is shown in Fig. 1. 1 0.5 0 -4 -2 0 2 4 -0.5 -1 x Figure 1: Gaussian function As always, it can be useful to draw pictures to help you think about integrals. The fundamental integral is Z +1 rπ exp −λx2 dx = (2) −∞ λ 1 You should memorize either this or the related integral (I prefer the first one) Z +1 1rπ exp −λx2 dx = (3) 0 2 λ The only difference between these two is the limits of integration. Always pay attention to the limits of integration. Always. Not paying attention to the limits of integration is a common source of mistakes. We can see this by drawing the second function and because integrals are 1 0.5 0 -4 -2 0 2 4 -0.5 -1 x Figure 2: areas under a curve it becomes obvious that Eq. 3 is half of Eq. 2. (As an aside, although you do not need to learn this derivation, this is how one can derive the basic Gaussian integral. Suppose we want Z +1 I = exp −λx2 dx: −∞ Then we square this: Z +1 Z +1 I2 = exp −λx2 dx exp −λy2 dy (4) −∞ −∞ which we rewrite as ZZ I2 = exp −λ(x2 + y2) dx dy: (5) Now we go from Cartesian coordinates (x; y) to polar coordinates (r; θ): ZZ I2 = exp −λr2 rdr dθ: (6) 2 The integral over θ is easy, leaving Z 1 I2 = 2π exp −λr2 rdr: (7) 0 Now we make a substitution: u = r2, with du = 2rdr or Z 1 1 I2 = 2π exp (−λu) du: (8) 0 2 But this is the simple integral of an exponential (you have to start with some sort of integral) and 1 2 1 π I = π × exp (−λu) = : (9) −λ 0 λ From this we immediately get I is Eq. -
![Arxiv:1812.07690V1 [Math.GT] 18 Dec 2018 Nahm Sums Are Special Q-Hypergeometric Series Whose Summand Involves a Quadratic Form, a Linear Form and a Constant](https://docslib.b-cdn.net/cover/6341/arxiv-1812-07690v1-math-gt-18-dec-2018-nahm-sums-are-special-q-hypergeometric-series-whose-summand-involves-a-quadratic-form-a-linear-form-and-a-constant-4386341.webp)
Arxiv:1812.07690V1 [Math.GT] 18 Dec 2018 Nahm Sums Are Special Q-Hypergeometric Series Whose Summand Involves a Quadratic Form, a Linear Form and a Constant
ASYMPTOTICS OF NAHM SUMS AT ROOTS OF UNITY STAVROS GAROUFALIDIS AND DON ZAGIER Abstract. We give a formula for the radial asymptotics to all orders of the special q- hypergeometric series known as Nahm sums at complex roots of unity. This result is used in [2] to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in K-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm sums that will be discussed further in a subsequent publication. Contents 1. Introduction 1 2. Asymptotic formula for the summand of a Nahm sum2 3. Asymptotic formula for a Nahm sum4 4. Proof of the asymptotic formulas6 4.1. Proof of Lemma 2.1 6 4.2. Proof of Proposition 2.27 4.3. Proof of Theorem 3.1 8 5. A syntactical identity among two collections of formal power series 10 6. Coefficient versus radial asymptotics of Nahm sums 12 7. Modular Nahm sums 13 Appendix A. Application: proof of the Kashaev-Mangazeev-Stroganov identity 14 References 16 1. Introduction arXiv:1812.07690v1 [math.GT] 18 Dec 2018 Nahm sums are special q-hypergeometric series whose summand involves a quadratic form, a linear form and a constant. They were introduced by Nahm [14] in connection with characters of rational conformal field theories. Nahm formulated a very surprising conjecture, that has elicited a lot of interest, relating the question of their modularity to the vanishing of a certain invariant in algebraic K-theory (more specifically, in K3-group, or equivalently the Bloch group, of the algebraic numbers). -
Poisson's Remarkable Calculation
Poisson’s remarkable calculation - a method or a trick? Denis Bell1 Department of Mathematics, University of North Florida 1 UNF Drive, Jacksonville, FL 32224, U. S. A. email: [email protected] x2 The Gaussian function e− plays a fundamental role in probability and statis- tics. For this reason, it is important to know the value of the integral ∞ x2 I = e− dx. Z0 Since the integrand does not have an elementary antiderivative, I cannot be evaluated directly by the fundamental theorem of calculus. The familiar com- putation of the Gaussian integral is via the following remarkable trick, attributed to Poisson. One forms the square of I, interprets it as a double integral in the plane, transforms to polar coordinates and the answer magically pops out. The calculation is as follows 2 ∞ ∞ (x2+y2) I = e− dxdy Z0 Z0 π/2 ∞ r2 π ∞ r2 = re− drdθ = re− dr 2 Z0 Z0 Z0 r2 ∞ πe− π = − = . 4 4 ,0 Hence I = √π/2. Can this method be used to compute other seemingly intractable integrals? In this article we explore this question. Consider an improper integral ∞ J f(x)dx ≡ Z0 and suppose that f satisfies the functional equation f(x)f(y) = g(x2 + y2)h(y/x), x, y > 0. (1) 1Research partially supported by NSF grant DMS-0451194 1 Poisson’s remarkable calculation 2 Then Poisson’s argument yields the relation π/2 1 ∞ J 2 = g(x)dx h(tan θ)dθ . (2) 2 0 0 ≥ Z ¥≥ Z ¥ The determination of J is thereby reduced to the evaluation of two, possibly solvable, integrals. -
The Integrals in Gradshteyn and Ryzhik. Part 7: Elementary Examples
THE INTEGRALS IN GRADSHTEYN AND RYZHIK. PART 7: ELEMENTARY EXAMPLES TEWODROS AMDEBERHAN AND VICTOR H. MOLL Abstract. The table of Gradshteyn and Ryzhik contains some elementary integrals. Some of them are derived here. 1. Introduction Elementary mathematics leave the impression that there is marked difference between the two branches of calculus. Differentiation is a subject that is systematic: every evaluation is a consequence of a number of rules and some basic examples. However, integration is a mixture of art and science. The successful evaluation of an integral depends on the right approach, the right change of variables or a patient search in a table of integrals. In fact, the theory of indefinite integrals of elementary functions is complete [3, 4]. Risch’s algorithm determines whether a given function has an antiderivative within a given class of functions. However, the theory of definite integrals is far from complete and there is no general theory available. The level of complexity in the evaluation of a definite integral is hard to predict as can be seen in 2 3 ∞ x ∞ x √π ∞ x 4 (1.1) e− dx =1, e− dx = , and e− dx =Γ . 2 3 Z0 Z0 Z0 The first integrand has an elementary primitive, the second one is the classical Gaussian integral and the evaluation of the third requires Euler’s gamma function defined by ∞ a 1 x (1.2) Γ(a)= x − e− dx. Z0 arXiv:0707.2122v1 [math.CA] 14 Jul 2007 The table of integrals [5] contains a large variety of integrals. This paper contin- ues the work initiated in [1, 7, 8, 9, 10, 11] with the objective of providing proofs and context of all the formulas in the table [5].