THE ERROR FUNCTION
9. THE ERROR FUNCTION
You will meet this function in probability theory (Chapter 15, Section 8), and consequently in statistical mechanics and other applications of probability theory. You have probably heard of “grading on a curve.” The “curve” means the bell-
shaped graph of y = e −x 2 (see Problem 1); the error function is the area under part of this curve. We define the error function as
erf(x) = √
e −t 2 dt.
Although this is the usual definition of erf(x), there are other closely related inte- grals which are used and sometimes referred to as the error function. Consequently, you need to look carefully at the definition in the reference you are using (text, tables, computer). Here are some integrals you may find and their relation to (9.1) (see Problem 2).
The standard normal or Gaussian cumulative distribution function Φ(x) [see Chapter 15, equation (8.5)]:
√ (9.2a)
Φ(x) = √
e −t 2 /2 dt = 1 + 1 2 erf(x/
e −t 2 /2 dt = 1 erf(x/
The complementary error function:
(9.3a) 2 erfc(x) = √ e −t dt = 1 − erf(x),
(9.3b) 2 erfc √ = e −t /2 dt.
We can also use (9.2) to write erf(x) in terms of the standard normal cumulative distribution function [Chapter 15, equation (8.5)].
erf(x) = 2Φ(x
548 Special Functions Chapter 11
We next consider several useful facts about the error function. You can easily prove that the error function is odd; that is, erf(−x) = − erf(x) (Problem 3). We show that erf(∞) = 1 as follows:
by (5.2) and (5.3). For very small values of x, erf(x) can be approximated by ex- −t panding e 2 in a power series and integrating term by term. We get
e −t 2 2 dt = 2 √
erf(x) = √
3 5 · 2! [Use this when |x| ≪ 1. Compare (10.4).]
For large x, say x > 3, erf(x) differs from erf(∞) = 1 [see (9.5)] by less than
10 −4 (and of course even less for larger x). We are then usually interested in
1 − erf(x) = erfc(x). This is best approximated by an asymptotic series; we shall discuss such expansions in Section 10. The function erfi(x), called the imaginary error function, is similar to the error function but with a positive exponential. We define
erfi(x) = √
e t 2 dt.
You can show (Problem 5) that erf(ix) = i erfi(x). The Fresnel integrals (Chapter 1, Section 15) are related to the error function (Problem 6). Also see Section 10, Problem 3 for other relations involving error functions.
PROBLEMS, SECTION 9
1. Sketch or computer plot a graph of the function y = e −x 2 . 2. Verify equations (9.2), (9.3), and (9.4). Hint: In (9.2a), you want to write Φ(x) √
in terms of an error function. Make the change of variable t = u 2 in the Φ(x) √ integral. Warning: Don’t forget to adjust the limits; when t = x, u = x/ 2.
3. Prove that erf(x) is an odd function of x. Hint: Put t = −s in (9.1).
4. Show that
e −y /2 dy = 2π
(a) by using (9.5) and (9.2a); (b)
by reducing it to a Γ function and using (5.3). 5. Replace x by ix in (9.1) and let t = iu to show that erf(ix) = i erfi(x), where erfi(x)
is defined in (9.7).
Section 10 Asymptotic Series 549
6. Assuming that x is real, show the following relation between the error function and the Fresnel integrals.
= (1 − i) 2 (cos u + i sin u 2 ) du.
Hint: In (9.1), make the change of variables t = 1−i √ u.
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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