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Special Functions Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering. The survey of special functions presented here is not complete – we focus only on functions which are needed in this class. We study how these functions are defined, their main properties and some applications. Chapter 5 SPECIAL FUNCTIONS 5.1 Heaviside Step Function 5.1 Heaviside Function (unit step function) The Heaviside step function has only two values: 0 and 1 with a jump at x = 0 : 0 x < 0 H ()x = (1) 1 x > 0 Graphically it can be shown as: > plot(Heaviside(x),x=-3..3,discont=true); H ( x) Shifting of the step function along the x-axis: 0 x < a H ()x − a = (2) 1 x < a > plot(Heaviside(x-2),x=-1..4,discont=true); H ( x2− ) filter function The filter function can be constructed in terms of the step function: 0 x < a F()()()x,a,b = H x − a − H x − b = 1 a < x < b (3) 0 x > b It cuts the values of functions to zero outside of the interval []a,b : > F(x,1,3):=Heaviside(x-1)-Heaviside(x-3); > plot(g(x)*F(x,1,3),x=-1..5,discont=true); g ( xFx,1,3)⋅ ( ) The Heaviside step function is used for the modeling of a sudden increase of some quantity in the system (for example, a unit voltage is suddenly introduced into an electric circuit) – we call this sudden increase a spontaneous source. Chapter 5 SPECIAL FUNCTIONS 5.2 Dirac Function (delta function) The Dirac delta function δ (x) is not a function in the traditional sense – it is rather a distribution – a linear operator defined by two properties. The first describes its values to be zero everywhere except at x = 0 where the value is infinite: ∞ x = 0 δ ()x = (4) 0 x ≠ 0 The second property provides the unit area under the graph of the delta function: b ∫δ ()x dx = 1 where a < 0 and b > 0 a The delta function is vanishingly narrow at x = 0 but nevertheless encloses a finite area. It is also known as the unit impulse function. The Dirac delta function can be treated as the limit of the sequence of the following functions: a) rectangular functions: H ( xh+ )(−− Hxh ) δ ()x== lim Sh () x lim h0→→ h0 2h b) Gauss distribution functions: x2 − 1 σ 2 δ ()x ==lim Gσ () x lim e σσ→→00σπ c) triangle functions: 0x<− h −x1 +−≤<hx0 h2 h δ (x) = limδ h (x) δh ()x = h→0 x1 +≤≤0xh h2 h 0xh> d) Cauchy density (distribution) functions: n δ ()x== lim Dn lim nn→∞ →∞ π ()1nx+ 22 e) sine functions: sin nx δ ()x = lim n→∞ πx Chapter 5 SPECIAL FUNCTIONS Properties 1) Extension of the interval of integration to all real numbers still keeps the unit area under the graph of the delta function: ∞ ∫δ ()x dx = 1 −∞ 2) The Dirac delta function is a generalized derivative of the Heaviside step function: dH (x) δ ()x = dx It can be obtained from the consideration of the integral from the definition of the delta function with variable upper limit. It is obvious, that x 0 x < 0 ∫δ ()t dt = = H (x ) −∞ 1 x > 0 Therefore, the step function is formally an antiderivative of the delta function which now can be interpreted as a derivative of a discontinuous function. 3) Shifting in x : ∞ x = a δ ()x − a = 0 x ≠ a a+c ∫δ ()x − a dx = 1, c > 0 a−c 4) Symmetry: δδ(−=x) ( x) δδ( x − aax) =−( ) 5) Derivatives: 1 δ ′()x = − δ ()x x The derivative can be defined as a limit of triangle functions and interpreted as a pure torque in mechanics. The higher order derivatives of the delta function are: k! δ ()k ()x = (− 1 )k δ ()x k1,2,...= xk 6) Scaling: 1 δ ()ax = δ ()x for a ≠ 0 a 7) There are some important properties of the delta function which reflect its application to other functions. If f (x) is continuous at x = a , then f (x)δ (x − a) = f (a)δ (x − a) c ∫ f ()(x δ x − a )dx = f ()a b < a < c b ∞ ∫ f ()(x δ x − a )dx = f ()a −∞ x ∫ f ()t δ (t − a )dt = f (a )H (x − a ) x0 ≤ a x0 Chapter 5 SPECIAL FUNCTIONS Integration with derivatives of the delta function (integration by parts): ∞∞ ∞ fxδδ′ xdxfx=− x f′′ x δ xdx =− f 0 ∫∫() () () ()−∞ () () () −∞ −∞ ∞∞ ∞ f xδδ′′′′′′′′ xdx=− f x x f x δ xdx =−−= f 0′ f 0 ∫∫() () () ()−∞ () () () () −∞ −∞ 8) Laplace transform: ∞ L{}δ ()x = ∫ e−sxδ ()x dx = 1 0 ∞ L{}δ ()x − a = ∫ e−sxδ ()x − a dx = e−as a > 0 0 9) Fourier transform: ∞ F{}δ ()x − a = ∫ e−iωxδ ()x − a dx = e−iaω a > 0 −∞ Applications The delta function is applied for modeling of impulse processes. For example, the unit volumetric heat source applied instantaneously at time t = 0 is described in the Heat Equation by the delta function: ∂u − k∇2u = δ ()t ∂t If the unit impulse source is located at the point r = r0 and releases all energy instantaneously at time t = t0 , then the Heat Equation has a source ∂u − k∇2u = δ ()()t − t δ r − r ∂t 0 0 Impulse models are used for calculation of the Green’s function for non- homogeneous DE. The other interpretation of the delta function δ ()t − t0 as a force applied instantaneously at time t = t0 yielding an impulse of unit magnitude. Example Consider IVP: unit impulse is imposed on a dynamical system initially at rest at t = 5 : y′′ + 9 y = δ (t − 5) initial conditions: y(0) = 0 y′(0) = 0 Solution: Apply the Laplace transform to the given initial value problem (use the property of the Laplace transform): s 2Y + 9Y = e−5s Solve the algebraic equation forY : e−5s Y = s2 + 9 The inverse Laplace transform yields a solution of IVP: 1 y()t = H ()()t − 5 sin 3 t − 5 3 The graph of the solution shows that the system was at rest until the time t = 5 , when an impulse force was applied yielding periodic oscillations. Chapter 5 SPECIAL FUNCTIONS 5.3 Sine Integral Function The sine integral function is defined by the formula: x sint Si( x ) = ∫ dt x ∈ ()− ∞,∞ (5) 0 t The integrand can be expanded in Taylor series and then integrated term by term yielding a series representation of the sine integral function: ∞ (− 1)n x2n+1 Si()x = ∑ (6) n=0 ()()2n + 1 2n + 1 ! Graphically it can be shown as: > plot(Si(x),x=-25..25); Si( x) The limiting values of the sine integral function are determined by the Dirichlet integral ∞ sinω π ∫ dω = 0 ω 2 which was obtained in Section 3.5 (Remark 3.7, p.227) as a particular case of the Fourier transformation of the step function. In Chapter 3, we discussed connection of Gibbs phenomena in the Fourier series approximations of functions with jumps and the properties of sine integral function. Chapter 5 SPECIAL FUNCTIONS 5.4 Error Function The error function is the integral of the Gauss density function x 2 2 erf ( x ) = ∫ e−t dt x ∈ ()− ∞,∞ (7) π 0 > plot(erf(x),x=-4..4); erf ( x ) The complimentary error function is defined as ∞ 2 2 erfc( x ) = 1 − erf ()x = ∫e−t dt (8) π x > plot(erfc(x),x=-4..4); erfc( x ) Series expansion of the error function: 2 ∞ (− 1)n x2n+1 erf ()x = ∑ π n=0 n! ()2n + 1 Derivatives of the error function: d 2 2 erf ()x = e −x dx π 2 d − 4x 2 erf ()x = e −x dx 2 π Chapter 5 SPECIAL FUNCTIONS 5.5 Gamma Function Definition The Gamma function appears in many integral or series representations of special functions. Gamma was introduced by Leonard Euler in 1729 who investigated the integral function 1 q ∫ x p ()1xdx− p,q∈ 0 which for natural values p,q∈ is equal to p!q! ()pq1!++ With some transformation of this integral and taking the limits, Euler ended up with the result 1 n ∫ ()−ln x dx=≡ n!Γ () n + 1 0 Later, the gamma function was defined by the improper integral which converges for all x except of 0 and negative integers (Euler, 1781): ∞ Γ ()x = ∫ e−tt x−1dt (9) 0 > plot(GAMMA(x),x=-5..4); Γ ()x Properties: a) Γ (x + 1) = xΓ (x) (10) ∞ Γ (x + 1) = ∫ e−tt ()x+1 −1dt 0 ∞ = ∫ e−tt xdt 0 ∞ = −∫t xde−t 0 ∞ ∞ = − t xe−t + e−t dt x (limt xe−t = 0 ) 0 ∫ t→∞ 0 ∞ = x∫ e−tt x−1dt 0 = xΓ (x) Chapter 5 SPECIAL FUNCTIONS b) When x = n is a natural number then Γ (n) = (n − 1)! n = 1,2,3,..
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