DOCSLIB.ORG

Lecture 17. Fresnel Diffraction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 17. Fresnel Diffraction Lecture 17. Fresnel Diffraction J. Scott Tyo OPTI 512R College of Optical Sciences University of Arizona Last Updated August 10, 2011 1. THE FRESNEL REGION In the last lecture we looked at Fraunhofer diffraction in the far-field of the aperture. Given that the aperture was limited by a circle of radius L1, we found that we are in the the Fraunhofer region when πL 2 z >> 1 . 12 λ (1) There are a number of qualitative ways to describe this far field. They include • The distance after which the spherical waves look locally planar • The distance after which the electromagnetic wavefronts become spherical • The distance after which the ratio of the electric and magnetic fields is η = μ/. • The distance after which the field strength falls off as 1/r. • The distance after which the shape of the field distribution is a function of angle of observation only and not distance z. Before we get to the Fraunhofer region, we have the Fresnel region where these particular conditions are not satisfied. Consider the illustration in Fig. 1. Very close to the aperture, the intensity is approximately identical to the aperture shape. Very far from the aperture, in the Fraunhofer zone, the intensity is described by the Fourier transform of the aperture opening, and remains constant in shape as we move further away. Between Figure 1. Conceptual diagram of Rayleigh-Sommerfeld diffraction. Figure 10.4 from Gaskill 1 fields u2222 (x ,y ,z ) y1 y Aperture Plane 2 x1 x2 L2 L1 z=z z=z1 2 z12 Figure 2. Geometry for Fresnel diffraction. the near-field and the far-field is a region known as the Fresnel region where the fields continuously change from having the shape of the aperture to having the shape of the Fourier transform. This change must happen slowly, and Maxwell’s equations must be satisfied at all times. Consider once again the Kirchoff approximation to the Rayleigh-Sommerfeld diffraction integral ∞ jkr z12 e 12 u2(x2,y2)= u1(x1,y1) dx1dy1. (2) −∞ jλr12 r12 As we did last lecture, we write the distance r12 as 2 2 x2 − x1 y2 − y1 r12 = z12 1+ + (3) z12 z12 As we move back toward the aperture from the Fraunhofer zone (z12 decreases), we will need to consider additional terms in the Taylor series approximation for the square root. Writing the first three terms of the Taylor series yields 2 2 x2−x1 y2−y1 kz12 1+ z + z 12 12 (4) k 2 2 k 2 2 2 ≈ kz12 + (x2 − x1) +(y2 − y1) − 3 (x2 − x1) +(y2 − y1) + ... 2z12 8z12 In the Fraunhofer zone we completely ignored the final term in Eq. 4 and even discarded the terms that varied 2 2 2 like (x1 + y1 )/z12 .Asz12 decreases, we will now search for conditions under which we can ignore the last term on the right-hand side of Eq. 4, which will produce the Fresnel condition. We have already limited our aperture to having a radius L1. If we now limit our observations to be contained inside a circle of radius L2 as shown in Fig. 2, we can say that 2 2 2 (x2 − x1) +(y2 − y1) ≤ (L1 + L2) . (5) If the last term is much less than one radian, we can ignore its effects on the exponential. This condition is 4 π(L1 + L2) z3 >> , (6) 12 4λ 2 3 Spherical Wavefront Parabolic Approximation 2 Planar Approximation 1 y 0 −1 −2 −3 0 1 2 3 4 x Figure 3. Parabolic (Fresnel) and planar (Fraunhofer) approximations to the Spherical wave. and we can write jkz12 ∞ e π 2 2 u2(x2,y2)= u1(x1,y1)exp j (x2 − x1) +(y2 − y1) dx1dy1. (7) jλz12 −∞ λz12 Equation 6 tells us when we are inside the Fresnel region. It is important to note that the Fresnel region extends all the way to ∞, which means that the Fraunhofer region is itself a part of the Fresnel region. It is also important to note that the Fresnel region is not necessarily in the far-field of the aperture. Furthermore, the beginning of the Fresnel region depends on the radius of the circle L2 that inscribes our desired observations. The Fresnel zone actually begins closer to the aperture for observation points near the axis than for observation points further away. Let’s take a moment to consider the Fresnel approximation to the Kirchoff diffraction integral. In the Kirchoff integral, we have the kernel exp{jkr12}. We know that this kernel gives us spherical wave fronts, and provides us with a spherical wave. In the Fresnel approximation, we replace π jkr12 2 2 e ≈ exp j (x2 − x1) +(y2 − y1) . (8) λz12 We see then that the Fresnel approximation takes a spherical wave and approximates it with quadratic phase curvature. When x1 =0andy1 = 0, that is, the point source is at the origin, we have the fields jkz e 12 j π (x 2+y 2) u0(x2,y2) ≈ e λz12 2 2 . (9) jλz12 The right hand side of Eq. 9 is then the quadratic phase curvature approximation (Fresnel approximation) to a spherical wave. Whereas the Fraunhofer zone was reached when the spherical wave fronts are approximately planar, the Fresnel zone begins when the spherical wave fronts are approximately parabolic, as shown in Fig. 3 From this point forward whenever we see the term j π x2+y2 u(x, y) ∝ e λR ( ), (10) we should recognize it as the quadratic approximation to a spherical wave with radius of curvature R in the plane of observation. 3 2. THE FRESNEL DIFFRACTION INTEGRAL Equation 7 is known as a Fresnel transform of the fields in the aperture given by u1(x1,y1). The Fresnel transform is like a Fourier transform, only more complicated because it involves terms of the form exp{j2πξx} and exp j2πx2. The transform can be computed using the Fresnel integrals, which are tabulated. However, even though they are tabulated, they are not included in some software packages (like Matlab, though they are included in other packages, like Mathematica). If we take another look at Eq. 7, we recognize the Fresnel transform as having the properties of a convolution integral u2(x, y)=u1(x, y) ∗ h12(x, y) (11) where we define the impulse response as jkz12 e π 2 2 h12(x, y)= exp jk (x + y ) . (12) jλz12 λz12 This impulse response has a tabulated Fourier transform in table 7-3 of Gaskill, and we have the transfer function 2 2 jkz12 −jπλz12(ξ +η ) H12(ξ,η)=e e . (13) This is our first hint that we will be able to consider optical propagation in terms of a LSI system, and consider both impulse responses – what we refer to as point-spread functions – and transfer functions in how images propagate. 2.1. Using Fourier Transforms to Evaluate Fresnel Diffraction If we now take the definition of the Fresnel transform and expand its terms, we can write jkz ∞ e 12 π 2 2 π 2 2 2π j (x +y ) j (x +y ) −j (x1x2+y1y2) u2(x2,y2)= e λz12 2 2 u1(x1,y1)e λz12 1 1 e λz12 dx1dy1 (14) jλz12 −∞ jkz e 12 j π (x 2+y 2) j π (x 2+y 2) e λz12 2 2 F u x ,y e λz12 1 1 . = 1( 1 1) x y (15) jλz12 ξ= 2 ,η= 2 λz12 λz12 Equation 15 tells us that the Fresnel diffraction pattern can be computed by taking the Fourier transform of the aperture fields modified by a phase curvature term in the aperture plane. Let’s consider the various pieces of this diffraction integral. We have a spherical curvature term in terms of the observation position (x2,y2) that precedes the integral. Inside the integrand, we have the linear Fourier transform kernel that we saw in Fraunhofer diffraction, but we also have a spherical phase curvature term in terms of the source position (x1,y1). When we get far enough away (z12 →∞), the spherical curvature term inside the Fourier transform can be ignored, and the Fresnel diffraction approximation reduces to the Fraunhofer approximation that we saw last time. Let’s consider the following example. Our aperture is taken to be circular with radius 1 cm. The wavelength is 1 μm. The Fresnel distance is computed to be 20 cm and the Fraunhofer distance is computed as 315 m. For numerical reasons, we compute the Fresnel diffraction pattern at three different distances that are well inside the Fresnel zone, and we see the fields evolve from a reasonably complicated distribution to the Fraunhofer pattern. Note that the distribution is approximately Fraunhofer at a distance of 50 m, even though this is well below the far-field limit. These results are presented in Fig. 4. 4 Figure 4. Fresnel diffraction patterns for a circular aperture with a =1[cm],λ =1[μm]. 2.2. Fresnel integrals and the Cornu Spiral In the previous section we saw how we could use the tools of the Fourier transform (typically evaluated numer- ically for realistic problems) to evaluate Fresnel diffraction integrals. In this section, we will look at the direct evaluation of the Fresnel transform through the Fresnel integrals. As an example evaluation, let’s consider the Fresnel diffraction from a rectangular aperture. The transmission function of the aperture is x y t x ,y 1 , 1 . ( 1 1) = rect a b (16) In this case, assuming normally-incident plane wave illumination, we can write the diffraction as jkz ∞ e 12 π 2 2 j (x2−x1) +(y2−y1) u2(x2,y2)= t(x1,y1)e λz12 dx1dy1 (17) jλz12 −∞ jkz ∞ ∞ e 12 x π 2 y π 2 1 j (x2−x1) 1 j (y2−y1) = rect e λz12 dx1 rect e λz12 dy1 (18) jλz12 −∞ a −∞ b ejkz12 = IxIy.
Load more

Recommended publications
  • Fresnel Integral Computation Techniques
    FRESNEL INTEGRAL COMPUTATION TECHNIQUES ALEXANDRU IONUT, , JAMES C. HATELEY Abstract. This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635{661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor series, modified trapezoid rule and an asymptotic expansion for small, medium and large arguments respectively. These special functions can be computed accurately and efficiently up to an arbitrary preci- sion. Error estimates are provided and we give a systematic method in choosing the various parameters for a desired precision. We illustrate this method and verify numerically using double precision. 1. Introduction The Fresnel integrals and their simultaneous parametric plot, the clothoid, have numerous applications including; but not limited to, optics and electromagnetic theory [8, 15, 16, 17, 21], robotics [2,7, 12, 13, 14], civil engineering [4,9, 20] and biology [18]. There have been numerous works over the past 70 years computing and numerically approximating Fresnel integrals. Boersma established approximations using the Lanczos tau-method [3] and Cody computed rational Chebyshev approx- imations using the Remes algorithm [5]. Another approach includes a spreadsheet computation by Mielenz [10]; which is based on successive improvements of known relational approximations. Mielenz also gives an improvement of his work [11], where the accuracy is less then 1:e-9. More recently, Alazah, Chandler-Wilde and LaPorte propose a method to compute these integrals via a modified trapezoid rule [1]. Alazah et al. remark after some experimentation that a truncation of the Taylor series is more efficient and accurate than their new method for a small argument [1].
    [Show full text]
  • Analytical Evaluation and Asymptotic Evaluation of Dawson's Integral And
    Analytical evaluation and asymptotic evaluation of Dawson’s integral and related functions in mathematical physics V. Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Abstract Dawson’s integral and related functions in mathematical physics that in- clude the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev’s in- tegral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane C are derived using the asymptotic expansion of the confluent hypergeometric function. Keywords: Dawson’s integral, Complex error function, Plasma dispersion function, Fresnel functions, Gordeyev’s integral, Confluent hypergeometric function, asymptotic expansion 1. Introduction Let us consider the first-order initial value problem, D′ +2zD =1,D(0) = 0. (1) arXiv:1703.06757v1 [math.CA] 14 Mar 2017 Its solution given by the definite integral z z2 η2 daw z = D(z)= e− e dη (2) Z0 Email address: [email protected] (V. Nijimbere) Preprint submitted to Elsevier March 21, 2017 is known as Dawson’s integral [1, 17, 24, 27]. Dawson’s integral is related to several important functions (in integral form) in mathematical physics that include Faddeeva’s integral (also know as the complex error function or Kramp function) [9, 10, 22, 18, 27] z z2 2i z2 z2 2i η2 w(z)= e− 1+ e daw z = e− 1+ e dη , (3) √π √π Z0 Fried-Conte function (or plasma dispersion function) [4, 11] z z2 2i z2 z2 2i η2 Z(z)= i√πw(z)= i√πe− 1+ e daw z = i√πe− 1+ e dη , √π √π Z0 (4) (Jackson) function [14] G(z)=1+ zZ(z)=1+ i√πzw(z) z2 2i z2 =1+ i√πze− 1+ e daw z √π z z2 2i η2 =1+ i√πze− 1+ e dη , (5) √π Z0 and Fresnel functions C(x) and S(x) [1] defined by the relation z iπz2 e 2 daw √iπz = eiπη dη = C(x)+ iS(x), (6) √iπ Z0 where z z C(x)= cos(πη2)dη and S(x)= sin (πη2)dη.
    [Show full text]
  • Oscillatority of Fresnel Integrals and Chirp-Like Functions
    D ifferential E quations & A pplications Volume 5, Number 4 (2013), 527–547 doi:10.7153/dea-05-31 OSCILLATORITY OF FRESNEL INTEGRALS AND CHIRP–LIKE FUNCTIONS MAJA RESMAN,DOMAGOJ VLAH AND VESNA Zˇ UPANOVIC´ Dedicated to Luka Korkut, upon his retirement (Communicated by Yuki Naito) Abstract. In this review article, we present results concerning fractal analysis of Fresnel and gen- eralized Fresnel integrals. The study is related to computation of box dimension and Minkowski content of spirals defined parametrically by Fresnel integrals, as well as computation of box di- mension of the graph of reflected component function which are chirp-like function. Also, we present some results about relationship between oscillatority of the graph of solution of differ- ential equation, and oscillatority of a trajectory of the corresponding system in the phase space. We are concentrated on a class of differential equations with chirp-like solutions, and also spiral behavior in the phase space. 1. Introduction As coauthors of Luka Korkut, we present here an overview of his scientific con- tribution in fractal analysis of Fresnel integrals and chirp-like functions. We cite the main results from joint articles of Korkut, Resman, Vlah, Zubrini´ˇ candZupanovi´ˇ c, [18, 19, 20, 21, 22, 23]. The articles are mostly based on qualitative theory of differen- tial equations. A standard technique of this theory is phase plane analysis. We study trajectories of the corresponding system of differential equations in the phase plane, instead of studying the graph of the solution of the equation directly. Our main interest is fractal analysis of behavior of the graph of oscillatory solution, and of a trajectory of the associated system.
    [Show full text]
  • Fast and Accurate $ G^ 1$ Fitting of Clothoid Curves
    Fast and accurate G1 fitting of clothoid curves ENRICO BERTOLAZZI and MARCO FREGO University of Trento, Italy A new effective solution to the problem of Hermite G1 interpolation with a clothoid curve is here proposed, that is a clothoid that interpolates two given points in a plane with assigned unit tangent vectors. The interpolation problem is a system of three nonlinear equations with multiple solutions which is difficult to solve also numerically. Here the solution of this system is reduced to the computation of the zeros of one single function in one variable. The location of the zero associated to the relevant solution is studied analytically: the interval containing the zero where the solution is proved to exists and to be unique is provided. A simple guess function allows to find that zero with very few iterations in all possible configurations. The computation of the clothoid curves and the solution algorithm call for the evaluation of Fresnel related integrals. Such integrals need asymptotic expansions near critical values to avoid loss of precision. This is necessary when, for example, the solution of the interpolation problem is close to a straight line or an arc of circle. A simple algorithm is presented for efficient computation of the asymptotic expansion. The reduction of the problem to a single nonlinear function in one variable and the use of asymptotic expansions make the present solution algorithm fast and robust. In particular a comparison with algorithms present in literature shows that the present algorithm requires less iterations. Moreover accuracy is maintained in all possible configurations while other algorithms have a loss of accuracy near the transition zones.
    [Show full text]
  • Short Course on Asymptotics Mathematics Summer REU Programs University of Illinois Summer 2015
    Short Course on Asymptotics Mathematics Summer REU Programs University of Illinois Summer 2015 A.J. Hildebrand 7/21/2015 2 Short Course on Asymptotics A.J. Hildebrand Contents Notations and Conventions 5 1 Introduction: What is asymptotics? 7 1.1 Exact formulas versus approximations and estimates . 7 1.2 A sampler of applications of asymptotics . 8 2 Asymptotic notations 11 2.1 Big Oh . 11 2.2 Small oh and asymptotic equivalence . 14 2.3 Working with Big Oh estimates . 15 2.4 A case study: Comparing (1 + 1=n)n with e . 18 2.5 Remarks and extensions . 20 3 Applications to probability 23 3.1 Probabilities in the birthday problem. 23 3.2 Poisson approximation to the binomial distribution. 25 3.3 The center of the binomial distribution . 27 3.4 Normal approximation to the binomial distribution . 30 4 Integrals 35 4.1 The error function integral . 35 4.2 The logarithmic integral . 37 4.3 The Fresnel integrals . 39 5 Sums 43 5.1 Euler's summation formula . 43 5.2 Harmonic numbers . 46 5.3 Proof of Stirling's formula with unspecified constant . 48 3 4 CONTENTS Short Course on Asymptotics A.J. Hildebrand Notations and Conventions R the set of real numbers C the set of complex numbers N the set of positive integers x; y; t; : : : real numbers h; k; n; m; : : : integers (usually positive) [x] the greatest integer ≤ x (floor function) fxg the fractional part of x, i.e., x − [x] p; pi; q; qi;::: primes P summation over all positive integers ≤ x n≤x P summation over all nonnegative integers ≤ x (i.e., including 0≤n≤x n = 0) P summation over all primes ≤ x p≤x Convention for empty sums and products: An empty sum (i.e., one in which the summation condition is never satisfied) is defined as 0; an empty product is defined as 1.
    [Show full text]
  • Orange Peels and Fresnel Integrals
    Orange Peels and Fresnel Integrals LAURENT BARTHOLDI AND ANDRE´ HENRIQUES ut the skin of an orange along a thin spiral of The same spiral is also used in civil engineering: it constant width (Fig. 1) and place it flat on a table provides optimal curvature for train tracks between a CC(Fig. 2). A natural breakfast question, for a mathe- straight run and an upcoming bend [4, §14.1.2]. A train that matician, is what shape the spiral peel will have when travels at constant speed and increases the curvature of its flattened out. We derive a formula that, for a given cut trajectory at a constant rate will naturally follow an arc of width, describes the corresponding spiral’s shape. the Euler spiral. For the analysis, we parametrize the spiral curve by a The review [2] describes the history of the Euler spiral constant speed trajectory, and express the curvature of the and its three independent discoveries. flattened-out spiral as a function of time. For the purpose of our mathematical treatment, we shall This is achieved by comparing a revolution of the spiral replace the orange by a sphere of radius one. The spiral on on the orange with a corresponding spiral on a cone tan- the sphere is taken to be of width 1/N, as in Fig. 5. The area gent to the surface of the orange (Fig. 3, left). Once we of the sphere is 4p, so the spiral has a length of roughly 4p know the curvature, we derive a differential equation for N.
    [Show full text]
  • A Table of Integrals of the Error Functions*
    JOURNAL OF RESEAR CH of the Natiana l Bureau of Standards - B. Mathematical Sciences Vol. 73B, No. 1, January-March 1969 A Table of Integrals of the Error Functions* Edward W. Ng** and Murray Geller** (October 23, 1968) This is a compendium of indefinite and definite integrals of products of the Error function with elementary or transcendental functions. A s'ubstantial portion of the results are ne w. Key Words: Astrophysics; atomic physics; Error functions; indefi nite integrals; special functions; statistical analysis. 1. Introduction Integrals of the error function occur in a great variety of applicati ons, us ualJy in problems involvin g multiple integration where the integrand contains expone ntials of the squares of the argu­ me nts. Examples of applications can be cited from atomic physics [16),1 astrophysics [13] , and statistical analysis [15]. This paper is an attempt to give a n up-to-date exhaustive tabulation of s uch integrals. All formulas for indefi nite integrals in sections 4.1, 4.2, 4.5, a nd 4.6 below were derived from integration by parts and c hecked by differe ntiation of the resulting expressions_ Section 4.3 and the second hali' of 4.5 cover all formulas give n in [7] , with omission of trivial duplications and with a number of additions; section 4.4 covers essentially formulas give n in [4], Vol. I, pp. 233-235. All these formulas have been re-derived and checked, eithe r from the integral representation or from the hype rgeometric series of the error function. Sections 4.7,4_8 and 4_9 ori ginated in a more varied way.
    [Show full text]
  • Generalized Fresnel Integrals
    Bull. Sci. math. 129 (2005) 1–23 www.elsevier.com/locate/bulsci Generalized Fresnel integrals S. Albeverio 1,2,∗, S. Mazzucchi 2 Institut für Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany Received 25 May 2004; accepted 26 May 2004 Available online 6 August 2004 Abstract A general class of (finite dimensional) oscillatory integrals with polynomially growing phase func- tions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic ex- pansion for “strong oscillations” is given. The expansion is in powers of h¯ 1/2M,whereh¯ is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions. 2004 Elsevier SAS. All rights reserved. Résumé On étudie une classe générale d’intégrales oscillatoires en dimension finie avec une fonction de phase à croissance polynomiale. Une formule de représentation du type Parseval est démontrée, ainsi qu’une formule donnant les intégrales au moyen de la continuation analytique d’intégrales ab- solument convergentes. On donne les développements asymptotiques de ces intégrales dans le cas d’ “oscillations rapides”. Ces développements sont en puissance de h¯ 1/2M,oùh¯ est un petit para- mètre et 2M est l’ordre de croissance de la fonction de phase. Sous des conditions additionelles sur les integrands on obtient la convergence, resp. la sommabilité au sens de Borel, des développements asymptotiques. 2004 Elsevier SAS. All rights reserved. MSC: 28C05; 35S30; 34E05; 40G10; 81S40 * Corresponding author.
    [Show full text]
  • Chapter S – Approximations of Special Functions
    s – Approximations of Special Functions Introduction – s Chapter s – Approximations of Special Functions 1. Scope of the Chapter This chapter provides functions for computing some of the most commonly required special functions of mathematics. See Chapter g01 for probability distribution functions and their inverses. 2. Accuracy The majority of the functions in this chapter evaluate real-valued functions of a single real variable x. The accuracy of the computed value is discussed in each function document, along the following lines. Let ∆ be the absolute error in the argument x,andδ be the relative error in x. If we ignore errors that arise in the argument by propagation of data errors etc. and consider only those errors that result from the fact that a real number is being represented in the computer in floating-point form with finite precision, then δ is bounded and this bound is independent of the magnitude of x; for example, on an n-digit machine |δ|≤10−n. (This of course implies that the absolute error ∆ = xδ is also bounded but the bound is now dependent on x.) Let E be the absolute error in the computed function value f(x), and be the relative error. Then E |f (x)|∆ E |xf (x)|δ |xf (x)/f(x)|δ. If possible, the function documents discuss the last of these relations, that is the propagation of relative error, in terms of the error amplification factor |xf (x)/f(x)|. But in some cases, such as near zeros of the function which cannot be extracted explicitly, absolute error in the result is the quantity of significance and here the factor |xf (x)| is described.
    [Show full text]
  • Chapter S Approximations of Special Functions Contents
    S – Approximations of Special Functions Chapter S Approximations of Special Functions Contents 1 Scope of the Chapter 2 2 Background to the Problems 2 2.1 Functions of a Single Real Argument .............................. 2 2.2 Approximations to Elliptic Integrals .............................. 4 2.3 Bessel and Airy Functions of a Complex Argument ...................... 5 3 Recommendations on Choice and Use of Available Routines 5 3.1 Elliptic Integrals ......................................... 5 3.2 Bessel and Airy Functions .................................... 6 4Index 6 5 Routines Withdrawn or Scheduled for Withdrawal 7 6 References 7 [NP3390/19/pdf] S.1 Introduction – S S – Approximations of Special Functions 1 Scope of the Chapter This chapter is concerned with the provision of some commonly occurring physical and mathematical functions. 2 Background to the Problems The majority of the routines in this chapter approximate real-valued functions of a single real argument, and the techniques involved are described in Section 2.1. In addition the chapter contains routines for elliptic integrals (see Section 2.2), Bessel and Airy functions of a complex argument (see Section 2.3), exponential of a complex argument, and complementary error function of a complex argument. 2.1 Functions of a Single Real Argument Most of the routines for functions of a single real argument have been based on truncated Chebyshev expansions. This method of approximation was adopted as a compromise between the conflicting requirements of efficiency and ease of implementation on many different machine ranges. For details of the reasons behind this choice and the production and testing procedures followed in constructing this chapter see Schonfelder [7].
    [Show full text]
  • The Riemann Integral
    Chapter 1 The Riemann Integral I know of some universities in England where the Lebesgue integral is taught in the first year of a mathematics degree instead of the Riemann integral, but I know of no universities in England where students learn the Lebesgue integral in the first year of a mathematics degree. (Ap- proximate quotation attributed to T. W. K¨orner) Let f :[a,b] R be a bounded (not necessarily continuous) function on a compact (closed,→ bounded) interval. We will define what it means for f to be b Riemann integrable on [a,b] and, in that case, define its Riemann integral a f. The integral of f on [a,b] is a real number whose geometrical interpretation is the signed area under the graph y = f(x) for a x b. This number is also calledR the definite integral of f. By integrating f over≤ an≤ interval [a, x] with varying right end-point, we get a function of x, called the indefinite integral of f. The most important result about integration is the fundamental theorem of calculus, which states that integration and differentiation are inverse operations in an appropriately understood sense. Among other things, this connection enables us to compute many integrals explicitly. Integrability is a less restrictive condition on a function than differentiabil- ity. Roughly speaking, integration makes functions smoother, while differentiation makes functions rougher. For example, the indefinite integral of every continuous function exists and is differentiable, whereas the derivative of a continuous function need not exist (and generally doesn’t). The Riemann integral is the simplest integral to define, and it allows one to integrate every continuous function as well as some not-too-badly discontinuous functions.
    [Show full text]
  • Automatic Computing Methods for Special Functions. Part IV. Complex Error Function, Fresnel Integrals, and Other Related Functions
    JOURNAL OF RESEARCH of the National Bureau of Standards Vol. 86. NO.6. November-December 1981 Automatic Computing Methods for Special Functions. Part IV. Complex Error Function, Fresnel Integrals, and Other Related Functions Irene A. Stegun* and Ruth Zucker* National Bureau of Standards, Washington, DC 20234 July 15. 1981 Accurate, efficient, automatic methods for computing the complex error function to any precision are detailed and implemented in an American Standard FORTRAN subroutine. A six significant figure table of erfc z, ell erfc z, and eEl erfc( -z) is included for z in polar coordinate form with the modulus of z ranging from 0 to 9. The argand diagram is given for erf z. Key words: Argand diagram; complex error function; continued fraction; Dawson's function; FORTRAN subroutine; Fresnel integrals; key values; line broadening function; plasma dispersion function; Voigt function. 1. Introduction In computing many of the functions of mathematical physics, for example, Fresnel integrals, Dawson's integral, Voigt function, plasma dispersion function, etc., difficulties are frequently encountered. Since these functions may be expressed in terms of the error function of complex argument, we have chosen this function for Part IV.l The major part of the coding of the power series, continued fraction and asymptotic expansion computations for complex arguments will carryover equally well for other functions. As Part I was devoted to the error function of a real variable, the probability function and other related functions, Part IV will only emphasize those functions and pitfalls due to complex arguments. While accuracy over the entire domain of definition remains our main concern, the methods employed ensure efficiency, portability and ease of programming and modification.
    [Show full text]