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MATH306 SUPPLEMENTARY MATERIAL A BRIEF INTRODUCTION TO BESSEL and RELATED SPECIAL FUNCTIONS Edmund Y. M. Chiang c Draft date December 1, 2008 Contents Contents i 1 Trigonometric and Gamma Functions 3 1.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Bessel Functions 9 2.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Basic Properties of Bessel Functions . . . . . . . . . . . . . . . . . . . 12 2.3.1 Zeros of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Recurrence Formulae for Jν . . . . . . . . . . . . . . . . . . . 12 2.3.3 Generating Function for Jn . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Lommel’s Polynomials . . . . . . . . . . . . . . . . . . . . . . 14 2.3.5 Bessel Functions of half-integer Orders . . . . . . . . . . . . . 14 2.3.6 Formulae for Lommel’s polynomials . . . . . . . . . . . . . . . 14 2.3.7 “Pythagoras’ Theorem” for Bessel Function . . . . . . . . . . 15 2.3.8 Orthogonality of the Lommel Polynomials . . . . . . . . . . . 15 2.4 Integral formaulae 2.5 Asymptotic Behaviours of Bessel Functions . . . . . . . . . . . . . . . 17 2.5.1 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Addition formulae . . . . . . . . . . . . . . . . . . . . . . . . . 18 Fourier-Bessel Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 i CONTENTS 3 An New Perspective 1 21 3.1 Hypergeometric Equations . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Confluent Hypergeometric Equations . . . . . . . . . . . . . . . . . . 23 3.3 A Definition of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 23 4 Physical Applications of Bessel Functions 27 5 Separation of Variables of Helmholtz Equations 31 5.1 What is not known? . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 CONTENTS Chapter 1 Trigonometric and Gamma Functions 1.1 Trigonometric Functions Pythagoras Theorem: sin2 x + cos2 x = 1 (1.1) holds for all real x. Addition Theorem: (1.2) sin(A + B) = sin A cos B + sin B cos A (1.3) cos(A + B) = cos A cos B − sin A sin B hold for all angles A, B. 1.2 Gamma Function We recall that (1.1) k! = k × (k − 1) × · · · 3 × 2 × 1. Euler was able to give a correct definition to k! when k is not a positive integer. He invented the Euler-Gamma function in the year 1729. That is, Z ∞ (1.2) Γ(x) = tx−1 e−t dt. 0 3 4 CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS So the integral will “converge” for all positive real x. Since Z ∞ (1.3) Γ(1) = e−t dt = 1, 0 so one can show (1.4) Γ(x + 1) = xΓ(x), and so for each positive integer n (1.5) Γ(n + 1) = n!. In fact, the infinite integral will not only “converge” for all positive real x but can be “analytically continued” into the whole complex plane C. We have Z ∞ (1.6) Γ(z) = tz−1 e−t dt, <(z) > 0, 0 which is analytic in C except at the simple poles at the negative integers including 0. So it is a meromorphic function. Euler worked out that (1.7) 1 2 != 1√ π. 2 So 1 11 9 7 5 3 1 5 != , , , !. 2 2 2 2 2 2 2 One can even compute negative factorial: (1.8) (1.9) 1.3 Γ(1/2) = (−1/2)(−3/2)(−5/2)(−19/2)(−11/2)Γ(−11/2). Chebyshev Polynomials Theorem 1.3.1 (De Moivre’s Theorem). Let z = r(cos θ + i sin θ) be a complex number. Then for every non-negative integer n, (1.1) z n = rn (cos nθ + i sin nθ). Proof It is clear that that the formula holds when n = 1. We assume it holds at the integer n. Then the addition formulae for the trigonometric theorem imply that z n+1 = rn+1 (cos θ + i sin θ)(cos nθ + i sin nθ) = rn+1 [cos(n + 1)θ + i sin(n + 1)θ]. Thus, the result follows by applying the principle of induction. 1.3. CHEBYSHEV POLYNOMIALS 5 Exercise 1.3.2. Justify the formula for negative integers n. Remark 1.3.1. De Moivre’s formula actually works for all real and complex n. See MATH304. Example 1.3.3 (Multiple angle formulae). Let n = 2 in De Movire’s formula yields z 2 = cos 2θ + i sin 2θ. (1.2) But (1.3) z 2 = (cos θ + i sin θ)2 = cos2 θ − sin2 θ + i(2 cos θ sin θ). Comparing the real and imaginary parts of (1.2) and (1.3) yields cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ and (1.4) sin 2θ = 2 cos θ sin θ. Similarly, we have (1.5) z 3 = cos 3θ + i sin 3θ. But (1.6) z 3 = (cos θ + i sin θ)3 = cos3 θ + 3 cos2 θ(i sin θ) + 3 cos θ(i sin θ)2 + (i sin θ)3 = cos3 θ − 3 cos θ sin2 θ + i(3 cos2 θ sin θ − sin3 θ). Comparing the real and imaginary parts of (1.5) and (1.6) yields cos 3θ = cos3 θ − 3 cos θ sin2 θ = 4 cos3 θ − 3 cos θ and sin 3θ = 3 cos2 θ sin θ − sin3 θ = 3 sin θ − 4 sin3 θ. 6 CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS More generally, we have (1.7) cos nθ + i sin nθ = z n = (cos θ + i sin θ)n n X n = cosn−k θ(i sin θ)k k k=0 n X n k = i cosn−k θ sink θ. k k=0 Equating the real parts of the last equation and noting that i2l = (−1)l give n n n n−2 2 cos nθ = cos θ − cos θ sin θ + cosn−4 θ sin4 θ + · · · 2 4 n [n/2] + (−1) (1.8) cosn−2[n/2] θ sin2[n/2] θ. 2[n/2] But each of the sine function above has even power. Thus the substitution of sin2 θ = 1 − cos2 θ in (1.8) yields [n/2] (1.9) cos nθ = X l=0 l X n n−2l k l cos θ (−1) cos2k θ . 2l k k=0 Finally, it is possible to re-write the summation formula as (1.10) cos nθ = [n/2] X k=0 (−1) k [n/2] X j=k n 2j j cosn−2k θ. k Writing θ = arccos x for 0 ≤ θ ≤ π, and Tn (x) = cos(n arccos x). We deduce from the above formula that Tn (x) is a polynomial in x of degree n. The polynomial Tn is called the Chebyshev1 polynomial of the first kind of degree n. The first few polynomials are given by T0 (x) = 1; T1 (x) = x; T2 (x) = 2x2 − 1; T3 (x) = 4x3 − 3x; T4 (x) = 8x4 − 8x2 + 1; T5 (x) = 16x5 − 20x3 + 5x; ······ 1 P. L. Chebyshev (1821–1894) Russian mathematician. The collection of the polynomials {Tn (x)}∞ 0 is a complete orthogonal polynomial set on [−1, 1] 1.3. CHEBYSHEV POLYNOMIALS 7 The set of Chebyshev polynomials {Tn (x)} can generate a L2 [−1, 1] space. That is, it is a Hilbert space. Interested reader should consult the handbook [1] and the book [5] about the Chebyshev polynomials by T. J. Rivlin. Our library has a copy of this book. 8 CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS Chapter 2 Bessel Functions 2.1 Power Series We define the Bessel function of first kind of order ν to be the complex function represented by the power series (2.1) Jν (z) = +∞ +∞ X X (−1)k ( 21 z)ν+2k (−1)k ( 21 )ν+2k 2k = zν z . Γ(ν + k + 1) k! Γ(ν + k + 1) k! k=0 k=0 Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gamma function defined by Z ∞ (2.2) Γ(z) = tz−1 e−t dt, <(z) > 0. 0 2.2 Bessel Equation A common introduction to the series representation for Bessel functions of first kind of order ν is to consider Frobenius method: Suppose y(z) = ∞ X ck z α+k , k=0 where α is a fixed parameter and ck are coefficients to be determined. Substituting the above series into the Bessel equation (2.1) z2 dy d2 y + x + (z 2 − ν 2 )y = 0. 2 dz dz 9 10 CHAPTER 2. BESSEL FUNCTIONS and separating the coefficients yields: 0= = 2 2d y z dz 2 ∞ X +x dy + (z 2 − ν 2 )y dz ck (α + k)(α + k − 1)z α+k + k=0 ∞ X ck (α + k)z α+k k=0 + (z 2 − ν 2 ) · ∞ X ck z α+k k=0 = ∞ X 2 ck [(α + k) − ν ]z k=0 α+k + ∞ X ck z α+k+2 k=0 = c0 (α2 − ν 2 )z α + ∞ X {ck [(α + k)2 − ν 2 ] + ck−2 }z α+k k=0 The first term on the right side of the above expression is c0 (α2 − ν 2 )z α and the remaining are (2.2) (2.3) c1 [(α + 1)2 − ν 2 ] + 0 c2 [(α + 2)2 − ν 2 ] + c0 c3 [(α + 3)2 − ν 2 ] + c1 ······ 2 2 ck [(α + k) − ν ] + ck−2 ······ = 0 = 0 = 0 = 0 Hence a series solution could exist only if α = ±ν. When k > 1, then we require ck [(α + k)2 − ν 2 ] + ck−2 = 0, and this determines ck in terms of ck−2 unless α − ν = −2ν or α + ν = 2ν is an integer. Suppose we discard these exceptional cases, then it follows from (2.2) that c1 = c3 = c5 = · · · = c2k+1 = · · · = 0. Thus we could express the coefficients c2k in terms of c2k = (−1)k c0 . (α − ν + 2)(α − ν + 4) · · · (α − ν + 2k) · (α + ν + 2)(α + ν + 4) · · · (α + ν + 2k) 2.2. BESSEL EQUATION 11 If we now choose α = ν, then we obtain ∞ h X c0 z ν 1 + (2.4) k=0 i (−1)k ( 12 z)2k . k!(ν + 1)(ν + 2) · · · (ν + k) Alternatively, if we choose α = −ν, then we obtain c00 z −ν (2.5) h 1+ ∞ X k=0 i (−1)k ( 12 z)2k . k!(−ν + 1)(−ν + 2) · · · (−ν + k) Since c0 and c00 are arbitrary, so we choose them to be (2.6) c0 = 1 2ν Γ(ν + 1) , c00 = 1 2−ν Γ(−ν + 1) so that the two series (2.4) and (2.5) can be written in the forms: (2.7) Jν (z) = +∞ X (−1)k ( 21 z)ν+2k , Γ(ν + k + 1) k! k=0 J−ν (z) = +∞ X (−1)k ( 21 z)−ν+2k Γ(−ν + k + 1) k! k=0 and both are called the Bessel function of order ν if the first kind. It can be shown that the Wronskian of Jν and J−ν is given by (G. N. Watson “A Treatise On The Theory Of Bessel Functions”, pp. 42–43): (2.8) W (Jν , J−ν ) = − 2 sin νπ . πz This shows that the Jν and J−ν forms a fundamental set of solutions when ν is not equal to an integer. In fact, when ν = n is an integer, then we can easily check that (2.9) J−n (z) = (−1)n Jn (z). Thus it requires an extra effort to find another linearly independent solution. It turns out a second linearly independent solution is given by (2.10) Yν (z) = Jν (z) cos νπ − J−ν (z) sin νπ when ν is not an integer. The case when ν is an integer n is defined by (2.11) Jν (z) cos νπ − J−ν (z) . ν→n sin νπ Yn (z) = lim 12 CHAPTER 2. BESSEL FUNCTIONS The Yν so defined is linearly independent with Jν for all values of ν. In particular, we obtain n−1 (2.12) (2.13) −1 X (n − k − 1)! z 2k−n Yn (z) = π k=0 k! 2 ∞ i z 1 X (−1)k (z/2)n+2k h 2 log − ψ(k + 1) − ψ(k + n + 1) + π k=0 k!(n + k)! 2 for | arg z| < π and n = 0, 1, 2, · · · with the understanding that we set the sum to be 0 when n = 0. Here the ψ(z) = Γ0 (z)/Γ(z). We note that the function is unbounded when z = 0. 2.3 Basic Properties of Bessel Functions The general reference for Bessel functions is G. N. Watson’s classic: “A Treatise on the Theory of Bessel Functions”, published by Cambridge University Press in 1922 [6]. 2.3.1 Zeros of Bessel Functions See A. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with formulas, graphs and mathematical tables”, 10th Edt., National Bureau of Standards, 1964. [1]. [3] and [6] 2.3.2 Recurrence Formulae for Jν We consider arbitrary complex ν. d (−1)k z 2ν+2k d ν z Jν (z) = dz dz 2ν+2k k!Γ(ν + k + 1) d (−1)k z 2ν−1+2k = dz 2ν−1+2k k!Γ(ν + k) = z ν Jν−1 (z). But the left side can be expanded and this yields (2.1) zJν0 (z) + νJν (z) = zJν−1 (z). 2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS Similarly, d −nu z Jν (z) = −z −ν Jν+1 (z). dz (2.2) and this yields zJν0 (z) − νJν (z) = −zJν+1 (z). (2.3) Substracting and adding the above recurrence formulae yield 2ν Jν (z) z Jν−1 (z) − Jν+1 (z) = 2Jν0 (z). Jν−1 (z) + Jν+1 (z) = (2.4) (2.5) 2.3.3 Generating Function for Jn Jacobi in 1836 gave (2.6) e 1 z(t− 1t ) 2 = +∞ X tk Jk (z). k=−∞ Many of the forumulae derived above can be obtained from this expression. (2.7) e 1 z(t− 1t ) 2 = +∞ X ck (z)tk k=−∞ for 0 < |t| < ∞. We multiply the power series zt (2.8) e2 =1+ (z/2)2 2 (z/2) t+ t + ··· 1! 1! and zt e− 2 = 1 − (2.9) (z/2) (z/2)2 2 t+ t − ··· 1! 1! Multiplying the two series and comparing the coefficients of tk yield (2.10) (2.11) cn (z) = Jn (z), n = 0, 1, · · · cn (z) = (−1)n J−n (z), n = −1, −2, · · · . Thus (2.12) e 1 z(t− 1t ) 2 = J0 (z) + +∞ X k=1 Jk [tk + (−1)k t−k ]. 13 14 CHAPTER 2. BESSEL FUNCTIONS 2.3.4 Lommel’s Polynomials Iterating the recurrence formula 2ν Jν (z) − Jν−1 z with respect to ν a number of times give (2.13) Jν+1 (z) = Jν+k (z) = P (1/z)Jν (z) − Q(1/z)Jν−1 . (2.14) Lommel (1871) [6, pp. 294–295] found that Jν+k (z) = Rk,ν (z)Jν (z) − Rk−1,ν+1 Jν−1 . (2.15) 2.3.5 Bessel Functions of half-integer Orders One can check that (2.16) J− 1 (z) = 2 2 12 cos z, πz J 1 (z) = 2 2 12 sin z. πz Moreover, r (2.17) n Jn+ 1 (z) = (−1) 2 2 n+ 1 d n sin z z 2 , π z dz z n = 0, 1, 2, · · · . Thus applying a recurrence formula and using the Lommel polynomials yield Jn+ 1 (z) = Rn,ν (z)J 1 (z) − Rn−1,ν+1 J− 1 (z) (2.18) 2 2 2 That is, we have (2.19) 2.3.6 2 12 2 12 sin z − Rn−1,ν+1 sin z. Jn+ 1 (z) = Rn,ν (z) 2 πz πz Formulae for Lommel’s polynomials For each fixed ν, the Lommel polynomials are given by [n/2] X (−1)k (n − k)!(ν)n−k 2 n−2k (2.20) Rn ν (z) = k!(n − 2k)!(ν)k z k=0 where the [x] means the largest integer not exceeding x. Lommel is a German mathematician who made a major contribution to Bessel functions around 1870s. 2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS 2.3.7 15 “Pythagoras’ Theorem” for Bessel Function These Lommel polynomials have remarkable properties. Since 2 12 2 12 cos z, J 1 (z) = sin z (2.21) J− 1 (z) = 2 2 πz πz and sin2 x + cos2 x = 1; we now have 2 2 Jn+ (z) = 2(−1)n 1 (z) + J −n− 1 (2.22) 2 2 R2n, 1 −n (z) 2 πz . That is, we have 2 Jn+ 1 (z) 2 (2.23) + 2 J−n− 1 (z) 2 n 2 X (2z)2n−2k (2n − k)!(2n − 2k)! . = πz k=0 [(n − k)!]2 k! A few special cases are 1. J 21 (z) + J−2 1 (z) = 2 2 2. J 23 (z) + J−2 3 (z) = 2 2 2 ; πz 2 πz 1+ 2 1+ 2 2 πz 2 4. J 27 (z) + J−2 7 (z) = 1+ 2 2 πz 3. J 25 (z) + J−2 5 (z) = 2.3.8 1 ; z2 3 9 + ; z2 z4 6 45 225 + + 6 z2 z4 z Orthogonality of the Lommel Polynomials Let us set [n/2] (2.24) hn, ν (z) = Rn, ν (1/z) = X (−1)k (n − k)!(ν)n−k z n−2k , k!(n − 2k)!(ν) 2 k k=0 then the set {hn ν (z) is called the modified Lommel polynomials. Since the Bessel functions Jν (z) satisfies a three-term recurrence relation, so the Lommel polynomials inherit this property: (2.25) 2z(n + ν)hn ν (z) = hn+1, ν (z) + hn−1, ν (z) 16 CHAPTER 2. BESSEL FUNCTIONS with initial conditions (2.26) h0, ν (z) = 1, h1, ν (z) = 2νz. If one start with a different set of initial conditions (2.27) h∗0, ν (z) = 0, h∗1, ν (z) = 2ν, then the sequence {h∗n, ν (z)} generated by the (2.25) is called the associated Lommel polynomials. It is known that a three-term recurrence relation for polynomials with the coefficients of as in (2.25) will generate a set of orthogonal polynomials on R +∞(−∞, +∞). That is, there is a probability measure α onb (−∞, +∞) with dα = 1 −∞ Theorem 2.3.1 (A. A. Markov, 1895). Suppose the set of {pn (z)}of orthogonal polynomials with its measure α supported on a bounded internal [a, b], then Z b dα(t) p∗n ν (z) = (2.28) lim n→∞ pn ν (z) a z −t holds uniformly for z 6∈ [a, b]. Since we know Theorem 2.3.2 (Hurwitz). The limit (2.29) (z/2)ν+n Rn, ν+1 (z) = Jν (z), n→∞ Γ(n + ν + 1) lim holds uniformly on compact subsets of C. So we have Theorem 2.3.3. For ν > 0, the polynomials R +∞{hn ν (z) are orthogonal with respect to a discrete measure αν normalized to have −∞ dα(t) = 1, and Z dα(t) Jν (1/z) (2.30) = 2ν . Jν−1 (1/z) R z −t One can use this formula to recover the measure α(t). Thus the Lommel “polynomials” was found to have very distinct properties to be complete in L2 (−1, +1). 2.4. INTEGRAL FORMAULAE 2.4 17 Integral formaulae (2.1) (z/2)ν Jν (z) = 1 Γ( 2 )Γ(ν + 12 ) Z 1 (1 − t2 )ν−1/2 cos zt dt, <(ν) > −1/2, | arg z| < π. <(ν) > −1/2, | arg z| < π, −1 Or equivalently, (2.2) Z π (z/2)ν cos(z cos θ) sin2ν θ dθ, Jν (z) = 1 1 Γ( 2 )Γ(ν + 2 ) 0 where t = cos θ. (1) Writing Hν (z) = Jν (z) + iYν (z). Then Theorem 2.4.1 (Hankel 1869). (2.3) Hν(1) (z) 2 12 ei[z−νπ/2−π/4] Z ∞·exp(iβ) iu ν− 12 −u ν− 12 = e u du, 1+ πz Γ(ν + 1/2) 0 2z where |β| < π/2. 2.5 Asymptotic Behaviours of Bessel Functions Expanding the integrand of Henkel’s contour integral by binomial expansion and after some careful analysis, we have Theorem 2.5.1. For −π < arg z < 2π, (2.1) Hν(1) (x) = 2 πx 21 e i(x− 12 νπ− 41 π) X p−1 ( 12 − ν)m ( 12 + ν)m (1) + Rp (x) , m m! (2ix) m=0 and for −2π < arg x < π, (2.2) Hν(2) (x) = 2 πx 12 e −i(x− 12 νπ− 14 π) X p−1 ( 12 − ν)m ( 12 + ν)m (2) + Rp (x) , (2ix)m m! m=0 where (2.3) Rp(1) (x) = O(x−p ) and Rp(2) (x) = O(x−p ), as x → +∞, uniformly in −π + δ < arg x < 2π − δ and −2π + δ < arg x < π − δ respectively. 18 CHAPTER 2. BESSEL FUNCTIONS The two expansions are valid simultaneously in −π < arg x < π. (1) (2) We thus have Jν (z) = 12 (Hν (z) + Hν (z)). So Theorem 2.5.2. For | arg z| < π, we have Jν (z) ∼ 2 πz 21 h ∞ 1 X (−1)k (ν, 2k) 1 cos z − νπ − π 2 4 k=0 (2z)2k ∞ 1 X (−1)k (ν, 2k + 1) i 1 − sin z − νπ − π 2 4 k=0 (2z)2k+1 2.5.1 Addition formulae Schläfli (1871) derived (2.4) Jν (z + t) = ∞ X Jν−k (t)Jk (z). k=−∞ Theorem 2.5.3 (Neumann (1867)). Let z, Z, R forms a triangle and let φ be the angle opposite to the side R, then (2.5) ∞ p X 2 2 J0 k Jk (Z)Jk (z) cos kφ, (Z + z − 2Zz cos φ) = k=0 where 0 = 1, k = 2 for k ≥ 1. We note that Z, z can assume complex values. There are generalizations to ν 6= 0. 2.6 Fourier-Bessel Series Theorem 2.6.1. Suppose the real function f (r) is piecewise continuous in (0, a) and of bounded variation in every [r1 , r2 ] ⊂ (0, a). Then if Z a √ (2.1) r|f (r)| dr < ∞, 0 2.6. FOURIER-BESSEL SERIES 19 then the Fouier-Bessel series (2.2) f (r) = ∞ X ck Jν xνk k=1 r a converges to f (r) at every continuity point of f (r) and to (2.3) 1 [f (r + 0) + f (r − 0)] 2 at every discontinuity point of f (r). Here Z 1 2 xf (x)Jν (λk x) dx (2.4) ck = 2 Jν (λk ) 0 is the Fourier-Bessel coefficients of f . The xνk are the zero of Jν (x) k = 0, 1, · · · . The discussion of the orthogonality of the Bessel functions is omitted. You may consult G. P. Tolstov’s “ Fourier Series”, Dover 1962. The situation here is very much like the classical Fourier series in terms of sine and cosine functions. 20 CHAPTER 2. BESSEL FUNCTIONS Chapter 3 An New Perspective 3.1 Hypergeometric Equations We shall base our consideration of functions on the finite complex plane C in this chapter. Let p(z) and q(z) be meromorphic functions defined in C. A point z0 belongs to C is called a regular singular point of the second order differential equation d2 f (z) df (z) + p(z) + q(z)f (z) = 0 2 dz dz (3.1) if (3.2) lim (z − z0 )2 q(z) = B lim (z − z0 )p(z) = A, z→z0 z→z0 both exist and being finite. We start by recalling that any second order linear equation with three regular singular points in the C can always be transformed ([2, pp. 73–75]) to the hypergeometric differential equation (due to Euler, Gauss and Riemann) (3.3) z(1 − z) dy d2 y + [c − (a + b + 1)z] − ab y = 0 2 dz dz which has regular singular points at 0, 1, ∞. It is known that the hypergeometric equation has a power series solution (3.4) 2 F1 +∞ X (a)k (b)k k a, b z = F [a, b; c; z] = z 2 1 c (c)k k! k=0 21 22 CHAPTER 3. AN NEW PERSPECTIVE which converges in |z| < 1. It is called the Gauss hypermetric series which depends on three parameters a, b, c. Here the notation is the standard Pochhammer notation (1891) (a)k = a · (a + 1) · (z + 2) · · · (a + k − 1), (ν)0 = 1 for each integer k. Theorem 3.1.1 (Euler 1769). Suppose <(c) > <(b) > 0, then we have (3.5) 2 F1 Z 1 Γ(c) a, b z = tb−1 (1 − t)c−b−1 (1 − xt)−a dt, c Γ(b)Γ(c − b) 0 defined in the cut-plane C\[1, +∞), arg t = arg(1 − t) = 0, and that (1 − xt)−a takes values in the principal branch (−π, π). Thus Euler’s integal representation of 2 F1 thus continues the function analytically into the cut-plane. Special cases of (3.4) include: ... In 1812 Gauss gave a complete set of 15 contiguous relations ... We note that one can write familiar functions in terms of p Fq : 1. 2. 3. 4. 5. − e = 0 F0 x ; − a −a (1 − x) = 1 F0 x ; − 1, 1 log(1 + z) = x 2 F1 −x ; 2 − x2 cos x = 0 F1 − 4 ; 1/2 1/2, 1/2 2 −1 sin x = x 2 F1 x . 3/2 x 3.2. CONFLUENT HYPERGEOMETRIC EQUATIONS 3.2 23 Confluent Hypergeometric Equations Writing the (3.1) in the variable x = z/b results in the equation (3.1) x(1 − x/b) d2 y dy + [c − (a + b + 1)x/b] − a y = 0 2 dx dx which has regular singular points at 0, b, ∞. Letting b tends to ∞ results in (3.2) x d2 y dy + (c − x) − a y = 0 2 dx dx which is called the confluent hypergeometric equation. But the equation has a regular singular point at 0 and an irregular singular point at ∞. Substituting z/b in (3.4) and letting b to infinity results in a series (3.3) +∞ X (a)k k Φ(a, c; z) := 1 F1 [a; c; z] = z . (c)k k! k=0 This series is called a Kummer series. We note that the Gauss hypergeometric series has radius of convergence 1, and after the above substitution of z/b results in a series that has radius of convergence equal to |b|. So the Kummer series has radius of convergence ∞ after taking the limit b → ∞. We call all solutions to the equation (3.2) confluent hypergeometric functions because they are obtained from the confluence of singularities of the hypergeometric equations. Further details can be found in [4], better known as the Bateman project volumn one. 3.3 A Definition of Bessel Functions We define the Bessel function of first kind of order ν to be the complex function represented by the power series (3.1) +∞ +∞ X X (−1)k ( 21 z)ν+2k (−1)k ( 21 )ν+2k 2k ν =z z . Jν (z) = Γ(ν + k + 1) k! Γ(ν + k + 1) k! k=0 k=0 Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gamma function defined by Z ∞ (3.2) Γ(z) = tz−1 e−t dt, <(z) > 0. 0 24 CHAPTER 3. AN NEW PERSPECTIVE It can be analytically continued onto the whole complex plane C, and has simple poles at the negative integers including 0. So it is a meromorphic function. In particular, Γ(k + 1) = k! = k(k − 1) · · · 2 · 1. So the Euler-Gamma function is a generalization of the factorial notation k!. Let us set a = ν + 12 , c = 2ν + 1 and replace z by 2iz in the Kummer series. That is, ∞ X (ν + 21 )k Φ(ν + 1/2, 2ν + 1; 2iz) = (2iz)k (2ν + 1)k k! k=0 ν + 12 2iz = 1 F2 2ν + 1 z 2 − iz = e 0 F1 − ν + 1 2 ∞ X (−1)k z 2k = eiz (ν + 1)k k! 22k k=0 = eiz ∞ X (−1)k Γ(ν + 1) z 2k Γ(ν + k + 1) k! 2 k=0 ∞ z −ν z ν X (−1)k z 2k · = e Γ(ν + 1) 2 2 k=0 Γ(ν + k + 1) k! 2 z −ν = eiz Γ(ν + 1) · Jν (z), 2 iz where we have applied Kummer’s second transformation a − 2 2x x (3.3) 4x = e 0 F1 1 F1 2a a + 21 in the third step above, and the identity (ν)k = Γ(ν + k)/Γ(ν) in the fourth step. Since the confluent hypergeometric function Φ(ν + 1/2, 2ν + 1; 2iz) satisfies the hypergeometric equation (3.4) z d2 y dy + (c − z) − a y = 0 2 dz dz with a = ν + 1/2 and c = 2ν + 1. Substituting eiz Γ(ν + 1) and simplfying lead to the equation (3.5) z2 d2 y dy + x + (z 2 − ν 2 )y = 0. 2 dz dz z −ν 2 · Jν (z) into the (3.4) 3.3. A DEFINITION OF BESSEL FUNCTIONS 25 The above derivation shows that the Bessel function of the first kind with order ν is a special case of the confluent hypergeometric functions with specifically chosen parameters. 26 CHAPTER 3. AN NEW PERSPECTIVE Chapter 4 Physical Applications of Bessel Functions We consider radial vibration of circular membrane. We assume that an elastic circular membrane (radius `) can vibrate and that the material has a uniform density. Let u(x, y, t) denote the displacement of the membrane at time t from its equilibrium position. We use polar coordinate in the xy−plane by the change of variables: (4.1) x = r cos θ, y = r sin θ. Then the corresponding equation (4.2) 2 ∂ 2u ∂ 2u 2 ∂ u = c + ∂t2 ∂x2 ∂y 2 can be transformed to the form: (4.3) 2 ∂ 2u 1 ∂ 2u 1 ∂ 2u 2 ∂ u = c + + . ∂t2 ∂r2 r ∂r r2 ∂θ2 Since the membrane has uniform density, so u = u(r, t) that is, it is independent of the θ. Thus we have 2 ∂ 2u 1 ∂ 2u 2 ∂ u (4.4) . = c + ∂t2 ∂r2 r ∂r The boundary condition is u(`, t) = 0, and the initial conditions take the form (4.5) ∂u(r, 0) = g(r). ∂t u(r, t) = f (r), Separation of variables method yields (4.6) u(r, t) = R(r)T (t), 27 28 CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS which satisfies the boundary condition u(`, t) = 0. Thus 1 RT 00 = c2 R00 T + R0 T . r (4.7) Hence T 00 R00 + (1/r)R0 + 2 ; = −λ2 , R cT (4.8) where the λ is a constant. Thus 1 R00 + R0 + λ2 R = 0 r T 00 + c2 λ2 T = 0. (4.9) (4.10) We notice that the first equation is Bessel equation with ν = 0. Thus its general solution is given by (4.11) R(r) = C1 J0 (λr) + C2 Y0 (λr). . Since Y0 is unbounded when r = 0, so C2 = 0. Thus the boundary condition implies that (4.12) J0 (λ`) = 0, implying that µ = λ` is a zero of J0 (µ). Setting C1 = 1, we obtain for each integer n = 1, 2, 3, · · · , (4.13) Rn (r) = J0 (λn r) = J0 ( µn r), ` where µn = λn ` is the n−th zero of J0 (µ). Thus we have un (r, t) = (An cos cλn t + Bn sin cλn t) · J0 (λn r), (4.14) for n = 1, 2, 3, · · · . Thus the general solution is given by (4.15) ∞ X n=1 un (r, t) = ∞ X (An cos cλn t + Bn sin cλn t) · J0 (λn r), n=1 and by (4.16) f (r) = u(r, 0) = ∞ X n=1 An · J0 (λn r), 29 and (4.17) ∞ X ∂u(r, 0) g(r) = Bn cλn · J0 (λn r). = ∂t t=0 n=1 Fourier-Bessel Series theory implies that Z ` 2 (4.18) An = 2 2 rf (r)J0 (λn r) dr, ` J1 (µn ) 0 (4.19) 2 Bn = 2 cλn ` J12 (µn ) Z ` rg(r)J0 (λn r) dr. 0 30 CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS Chapter 5 Separation of Variables of Helmholtz Equations The Helmholtz equation named after the German physicst Hermann von Helmholtz refers to second order (elliptic) partial differential equations of the form: (5.1) (∆2 + k 2 )Φ = 0, where k is a constant. If k = 0, then it reduces to the Laplace equations. In this discussion, we shall restrict ourselves in the Euclidean space R3 . One of the most powerful theories developed in solving linear PDEs is the the method of separation of variables. For example, the wave equation 1 ∂2 (5.2) ∆2 − 2 2 Ψ(r, t) = 0, c ∂t can be solved by assuming Ψ(r, t) = Φ(r) · T (t) where T (t) = eiωt . This yields ω2 (5.3) ∆2 − 2 Φ(r) = 0, c which is a Helmholtz equation. The questions now is under what 3−dimensional coordinate system (u1 , u2 , u3 ) do we have a solution that is in the separation of variables form (5.4) Φ(r) = Φ1 (u1 ) · Φ2 (u2 ) · Φ3 (u3 ) ? Eisenhart, L. P. (”Separable Systems of Stäckel.” Ann. Math. 35, 284-305, 1934) determines via a certain Stäckel determinant is fulfilled (see e.g. Morse, P. M. and Feshbach, H. “Methods of Theoretical Physics, Part I”. New York: McGraw-Hill, pp. 125–126, 271, and 509–510, 1953). 31 32CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS Theorem 5.0.1 (Eisenhart 1934). There are a total of eleven curvilinear coordinate systems in which the Helmholtz equation separates. Each of the curvilinear coordinate is characterized by quadrics. That is, surfaces defined by (5.5) Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz + Gx + Hy + lz + J = 0. One can visit http://en.wikipedia.org/wiki/Quadric for some of the quadric surfaces. Curvilinear coordinate systems are formed by putting relevant orthogonal quadric surfaces. Wikipedia contains quite a few of these pictures. We list the eleven coordinate systems here: 33 Name of system (1) Cartesian Co-ordinate Systems Transformation formulae x = x, y = y, z = z Degenerate Surfaces (2) Cylindrical x = ρ cos φ, y = ρ sin φ, z = z ρ ≥ 0, −π < φ ≤ π (3) Spherical polar x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ r ≥ 0, 0 ≤ θ ≤ π, −π ≤ φ ≤ π Parabolic (4) cylinder x = u2 − v 2 , y = 2uv, z=z u ≥ 0, −∞ < v < +∞ Elliptic (5) cylinder x = f cosh ξ cos η, y = f sinh ξ sin η, z=z ξ ≥ 0, −∞ < η < +∞ Rotation (6) paraboloidal x = 2uv cos φ, 2uv sin φ, z = u2 − v 2 u, v ≥ 0, −π < φ < π Prolate (7) spheroidal x = ` sinh u sin v cos φ, y = ` sinh u sin v sin φ, z = ` cosh u cos v, u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π Finite line ; segment Two half-lines Oblate (8) spheroidal x = ` cosh u sin v cos φ, y = ` cosh u sin v sin φ, z = ` sinh u cos v, u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π Circular plate (disc); Plane with circular aperture (9) Paraboloidal x = 12 `(cosh 2α + 2 cos 2β − cosh 2γ, y = 2` cosh α cos β sinh γ, z = 2` sinh α sin β cosh γ, α, γ ≥ 0, −π < β ≤ π Parabolic plate; Plane with parabolic aperture (10) Elliptic conal x = krsn αsn β; y = (ik/k 0 )rcn αcn β, z = (1/k 0 )rdn αdn β; r ≥ 0, −2K < α ≤ 2K, β = K + iu, 0 ≤ u ≤ 2K 0 Plane sector; Including quarter plane 2 Half-plane Infinite strip; Plane with straight aperture Half-line 34CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS Coordinate system Laplace & Helmholtz Laplace Equation Helmholtz equation (1) Cartesian (Trivial) (Trivial) (2) Cylinderical (Trivial) Bessel (3) Spherical polar Associated Legender Associated Legender (4) Parabolic cylinder (Trivial) Weber (5) Elliptic cylinder (Trivial) Mathieu (6) Rotation-paraboloidal Bessel Confluent hypergeometric (7) Prolate spheroidal Associated Legender Spheroidal wave (8) Prolate spheroidal Associated Legender Spheroidal wave (9) Paraboloidal Mathieu Whittaker-Hill (10) Elliptic conal Lame Spherical Bessel, Lame (11) Ellipsoidal Lame Ellipsoidal 35 1. Associated Legender: (5.6) d2 y m2 o dy n (1 − x ) 2 − 2x + n(n + 1) − y=0 dx dx (1 − x2 ) 2 2. Bessel: x2 (5.7) d2 y dy + (x2 − n2 )y = 0 + x 2 dx dx 3. Spherical Bessel: x2 (5.8) d2 y dy + 2x + (x2 − n(n + 1))y = 0 2 dx dx 4. Weber: d2 y 1 + (λ − x2 )y = 0 2 dx 4 (5.9) 5. Confluent hypergeometric: (5.10) x dy d2 y + (γ − x) − αy = 0 2 dx dx 6. Mathieu: d2 y + (λ − 2q cos 2x)y = 0 dx2 (5.11) 7. Spheroidal wave: (5.12) o d2 y dy n µ2 2 2 (1 − x ) 2 − 2x + λ − + γ (1 − x ) y = 0 dx dx (1 − x2 ) 2 8. Lame: (5.13) d2 y + (h − n(n + 1)k 2 sn 2 x)y = 0 2 dx 9. Whittaker-Hill: (5.14) d2 y + (a + b cos 2x + c cos 4x)y = 0 dx2 10. Ellipsoidal wave: (5.15) d2 y + (a + bk 2 sn 2 x + qk 4 sn 4 x)y = 0 dx2 Remark 5.0.1. The spheroidal wave and the Whittaker-Hill do not belong to the hypergeometric equation regime, but to the Heun equation regime (which has four regular singular points). 36CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS 5.1 What is not known? It is generally regarded that the Bessel functions, Weber functions, Legendre functions are better understood, but the remaining equations/functions are not so well understood. 1. Bessel functions. OK! Still some unknowns. 2. Confluent hypergeometric equations/functions. NOT OK. 3. Spheroidal wave, Mathieu, Lame, Whittaker-Hill, Ellipsoidal wave are poorly understood. Some of them are relatd to the Heun equation which has four regular singular points. Its research has barely started despite the fact that it has been around since 1910. 4. Mathematicans are separating variables of Laplace/Helmholtz equations, but in more complicated setting (such as in Riemannian spaces, etc) Bibliography [1] Milton Abramowitz and Irene A. Stegun, editors. Handbook of mathematical functions, with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. Superintendent of Documents, US Government Printing Office, Washington, DC, 1965. Third printing, with corrections. [2] G. E. Andrews, R. A. Askey, and R. Roy. Special Functions. Cambridge University Press, Cambridge, 1999. [3] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume 2. McGraw-Hill, New York, 1953. [4] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume 1. McGraw-Hill, New York, 1953. [5] T. J. Rivlin and M. W. Wilson. An optimal property of Chebyshev expansions. J. Approximation Theory, 2:312–317, 1969. [6] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, second edition, 1944. 37