Solving Differential Equations in Terms of Bessel Functions

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16 CHAPTER 1. PRELIMINARIES The solutions in the case 0F1 can also be expressed in terms of Kummer functions with the Kummer formula: exp − x α 1F1 2 2α x − = x 0F1 1 2 + α 2 . 16 Concluding, we can separate the hypergeometric functions that satisfy a second order differential equation, into two classes. The 2F1 functions with three regular singularities and the Kummer functions with one regular and one irregular singularity. The 0F1-functions are a special case of the Kummer functions. They are close to the Bessel functions, which we will deal with in the following section. 1.3.2 Bessel Functions Definition 1.19 The solutions V (LB1) of the operator LB1 := x 2 ∂ 2 + x∂ + (x 2 − ν 2 ) (1.11) with the constant parameter ν ∈ C are called Bessel functions. The two linearly independent solutions Jν(x) := x ν ∞ 2 ∑ k=0 (−1) k k!Γ (ν + k + 1) and Yν(x) := Jν(x)cos(νπ) − J−ν(x) sin(νπ) x 2k 2 (1.12) (1.13) generate V (LB1) and they are called Bessel functions of first and second kind respectively. Similarly the solutions of LB2 := x 2 ∂ 2 + x∂ − (x 2 + ν 2 ) (1.14) are called the modified Bessel functions of first and second kind and they are generated by x ν ∞ 1 x 2k Iν(x) := 2 ∑ k=0 k!Γ (ν + k + 1) 2 −νπi πi =exp Jν exp x 2 2 and Kν(x) := π(I−ν(x) − Iν(x)) 2sin(νπ) (1.15) (1.16)
1.3. HYPERGEOMETRIC SERIES 17 Remarks 1.20 1. If 2ν /∈ Z, then Jν(x) and J−ν(x) also generate V (LB1). This is proven in [24, 7.2]. 2. In terms of hypergeometric functions, the Bessel functions of the first kind are given by x ν 1 − Jν(x) = 1 − (1.17) and Iν(x) = 2 x 2 Γ (ν + 1) 0F1 ν 1 Γ (ν + 1) 0F1 ν + 1 − ν + 1 4 x2 1 4 x2 . (1.18) Lemma 1.21 Replacing x by ix where i = √ −1 reduces LB2 to LB1 and vice versa. Proof. That this is true for Bessel functions of the first kind can easily be seen when comparing the hypergeometric representations (1.17) and (1.18). There we get Iν(x) = iνJν(ix), so there only remains a constant coefficient iν . That the statement for the corresponding operators is also true can be seen as follows. Let y(x) be a solution of LB2 and consider g = g(x) = y(ix), then g ′ (x) = i d dx y(x) x=ix and g ′′ (x) = − d2 y(x) = dx2 1 (ix) 2 ix d dx y(x) − ((ix) 2 + ν 2 )y(ix) . x=ix x=ix A general differential operator for g(x) is L = ∂ 2 + a1∂ + a0. Using the equations for g ′ and g ′′ we can transform Lg = 0 into 1 d + ia1 ix dx y(x) x=ix −1 + x2 (x2 − ν 2 ) + a + 0 y(ix) = 0. If we equate coefficients we obtain a1 = 1/x and a0 = (x 2 − ν 2 )/x 2 . Then L = 1/x 2 LB1 and g(x) ∈ V (L) = V (LB1). So g(x) is a solution of LB1. The reverse works similarly. Since LB1 and LB2 are so closely connected we just need to consider one of the two in later issues. From now on LB will refer to the modified Bessel operator LB2. The modified Bessel operator is easier to handle in some cases, as we will see in Example 1.32. There are various recurrence equations and relationships for the Bessel function and its derivative.
Page 1 and 2: Solving Differential Equations in T
Page 3 and 4: Contents List of Notations v Introd
Page 5 and 6: List of Notations This is a list of
Page 7 and 8: Introduction Ordinary differential
Page 9 and 10: 1 Preliminaries We will first intro
Page 11 and 12: 1.2. SINGULAR POINTS 11 A linear di
Page 13 and 14: 1.3. HYPERGEOMETRIC SERIES 13 Theor
Page 15: 1.3. HYPERGEOMETRIC SERIES 15 1.3.1
Page 19 and 20: 1.4. FORMAL SOLUTIONS AND GENERALIZ
Page 27 and 28: 2 Transformations From now on we wi
Page 29 and 30: 2.1. OPERATORS OF DEGREE TWO 29 in
Page 31 and 32: 2.1. OPERATORS OF DEGREE TWO 31 r 3
Page 33 and 34: 2.1. OPERATORS OF DEGREE TWO 33 The
Page 35 and 36: 2.1. OPERATORS OF DEGREE TWO 35 The
Page 37 and 38: 2.2. THE EXPONENT DIFFERENCE 37 > g
Page 39 and 40: 2.2. THE EXPONENT DIFFERENCE 39 whe
Page 41 and 42: 2.3. EQUIVALENCE OF DIFFERENTIAL OP
Page 47 and 48: 3 Solving in Terms of Bessel Functi
Page 49 and 50: 3.1. CHANGE OF VARIABLES 49 We star
Page 51 and 52: 3.1. CHANGE OF VARIABLES 51 ⇒ ∆
Page 53 and 54: 3.2. FINDING THE PARAMETER ν 53 Ex
Page 55 and 56: 3.2. FINDING THE PARAMETER ν 55 If
Page 57 and 58: 3.3. THE ALGORITHM 57 3.3 The Algor
Page 59 and 60: 3.3. THE ALGORITHM 59 Example 3.12
Page 61 and 62: 3.3. THE ALGORITHM 61 > gen exp(L,t
Page 63 and 64: 3.3. THE ALGORITHM 63 We can also f
Page 65 and 66: 3.3. THE ALGORITHM 65 > equiv(L,M);
Page 67 and 68:
3.3. THE ALGORITHM 67 must be a mul
Page 69 and 70:
3.3. THE ALGORITHM 69 2. Let ai,bi
Page 71 and 72:
3.3. THE ALGORITHM 71 The exp-regul
Page 73 and 74:
3.4. SOLVING OVER A GENERAL FIELD K
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
3.5. WHITTAKER FUNCTIONS 81 At x =
Page 83 and 84:
3.6. TWO FINAL EXAMPLES 83 > LW:=D
Page 85 and 86:
3.6. TWO FINAL EXAMPLES 85 Since Wh
Page 87 and 88:
4 Conclusion We developed an algori
Page 89 and 90:
A Appendix A.1 Transformations From
Page 91 and 92:
A.3. PACKAGE DESCRIPTION 91 coeffic
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A.3. PACKAGE DESCRIPTION 93 findWhi
Page 95 and 96:
Bibliography [1] ABRAMOV, S. A., BA
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BIBLIOGRAPHY 97 [23] VAN HOEIJ, M.
Page 99 and 100:
Index adjoint operator, 42, 44 alge
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Solving Differential Equations in Terms of Bessel Functions

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