Solving Differential Equations in Terms of Bessel Functions

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18 CHAPTER 1. PRELIMINARIES Lemma 1.22 The Bessel functions satisfy Jν+1(x) = 2ν x Jν(x) − Jν−1(x), J ′ ν(x) = ν x Jν(x) − Jν+1(x) (1.19) and similarly the modified Bessel functions satisfy Iν+1(x) = Iν−1(x) − 2ν x Iν(x), I ′ ν(x) = ν x Iν(x) + Iν+1(x). (1.20) Moreover, Yν(x) satisfies the same equations as Jν(x) and (−1) ν Kν(x) satisfies the same equations as Iν(x). Proof. [2, Equations 9.1.27 and 9.6.26]. The following result will be important to find solutions of differential operators. Corollary 1.23 Consider S := C(x)Bν + C(x)B ′ ν (1.21) where B ′ ν = d dx Bν and Bν is a linear combination of either Jν and Yν or Iν and (−1) ν Kν. The space S is invariant under the substitution ν → ν + 1. Proof. It follows from the last lemma that this is true for each Bessel function on its own. A linear combination of Jν and Yν does no harm since they satisfy the same equations. The same holds for a linear combination of Iν and (−1) ν Kν. 1.4 Formal Solutions and Generalized Exponents In this section we will study differential operators with power series coefficients in K = C((x)). In this context the derivation that is usually used is δ = x d dx . Definition 1.24 A universal extension U of K is a minimal (simple) differential ring in which every operator L ∈ K[∂] has precisely deg(L) C-linear independent solutions. Theorem 1.25 The universal extension U of K is unique and has the form U = K[{x a }a∈M,{e(q)}q∈Q,l], (1.22) where M ⊂ C is such that M ⊕ Q = C and Q := ∪m≥1x −1/m C[[x −1/m ]]. Here K denotes an algebraic closure of K and the following rules hold:
1.4. FORMAL SOLUTIONS AND GENERALIZED EXPONENTS 19 (i) The only relations between the symbols are x 0 = 1,x a+b = x a x b ,e(0) = 1 and e(q1 + q2) = e(q1)e(q2). (ii) The differentiation is given by δx a = ax a ,δe(q) = qe(q) and δl = 1. Proof. To give a complete proof we would have to introduce to many details about differential rings. This is why we sketch the idea here only and refer to [22, Chapter 3.2] for more details. In order to get an intuitive idea of the structure above, we will use the following statement from [22]: To get a differential ring where all equations y ′ = Ay with a matrix A over K have a fundamental system it is sufficient that all equations δy = ay, a ∈ K (1.23) and δy = 1 have a solution. Hence, to give a structure of U, we need to classify order one differential equations of the form (1.23) and define a solution of each of them in U. Doing this, we need to take care about equations that have the same solution. The function exp( a xdx) is a solution of (1.23). So two equations δy = ay and δy = by with a,b ∈ ¯k have the same solutions if exp( a xdx) = cexp( b xdx) for some constant δ f c ∈ K. This is exactly the case, if b = a + f for some f ∈ K, f = 0. δ f Thus, if I := { f | f ∈ K, f = 0}, then K/I gives us a classification of all the equations δy = ay with different solutions. We know that the algebraic closure consists of series with fractional exponents, i.e. K = C((x)) = ∪n∈NC((x1/n )). The degree of the lowest monomial in f ∈ K and δ f are the same. Thus, taking the quotient will give a series starting with a rational constant and we get the representation I = c + ∞ ∑ k=1 ckz k/n c ∈ Q,ck ∈ C,n ∈ N Then, of course, the elements in K/I are series in K which have a non-rational constant term. In Theorem 1.25 this was denoted by M ⊕ Q. We conclude that it is sufficient if the universal extension U has solutions of (1.23) for all a ∈ M and for all a ∈ Q. These are the elements {x a }a∈M and {e(q)}q∈Q respectively and finally, the symbol l is the solution to δy = 1. The interpretation of the symbols in the universal extension is the following: 1. x a is the function exp(aln(x)), 2. l is the function ln(x), and .
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 and 16: 1.3. HYPERGEOMETRIC SERIES 15 1.3.1
Page 17: 1.3. HYPERGEOMETRIC SERIES 17 Remar
Page 21 and 22: 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
Page 93 and 94:
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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