Solving Differential Equations in Terms of Bessel Functions

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vi LIST OF NOTATIONS Homk(k1, ¯k) embeddings of k1 in ¯k with fixed k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 K field of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 k field of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ¯k algebraic closure of k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 k[x] ring of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 K[∂] ring of differential operators with coefficients in K . . . . . . . . . . . . . 11 k(x) field of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 k[[x]] ring of finite power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 k((x)) field of infinite power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 LB modified Bessel operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 LCLM least common left multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 minpol(p) minimal polynomial of p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 N possibilities for the Bessel parameter ν . . . . . . . . . . . . . . . . . . . . . . . . 58 N(a) norm of an element a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 N(p) possibilities for ν corresponding to p ∈ N . . . . . . . . . . . . . . . . . . . . . 64 Ns possibilities for ν corresponding to s ∈ Sreg . . . . . . . . . . . . . . . . . . . . 58 ν parameter of the Bessel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 numer( f ) numerator of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 pFq rx Sirr Sreg tp hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ramification index of x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 set of exp-irregular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 set of exp-regular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 local parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Tr(a) trace of an element a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 V (L) solution space of an operator L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Introduction Ordinary differential equations have always been of interest since they occur in many applications. Yet, there is no general algorithm solving every equation. There have been developed various methods for different classes of differential equations and functions. Apart from those there are methods using symmetry properties, the computation of integrating factors and there certainly are many more. A special class of ordinary differential equations is the class of linear differential equations Ly = 0, for a linear differential operator L = n ∑ ai∂ i=0 i with coefficients in some differential field K, e.g. K = Q(x) and ∂ = d dx . The algebraic properties of those operators and their solutions spaces are studied very well, e.g. in [22]. Solutions that correspond to an order one right factor can always be found by Beke’s algorithm but also with algorithms described in [8] and [18]. Each of those factors corresponds to a hyperexponential solution. This is a solution of the form exp( r) for a rational function r. In general, one can also factor L into factors of lower degree [23]. From this point on, one will have to consider special functions, which are functions defined by a differential operator. The question of solving an equating in terms of a special function is equivalent to the question whether two differential operators can be transformed into each other by certain transformations. We will consider a change of variables x → f , exp-products y → exp( r)y and gauge transformations y → r0y + r1y ′ . The parameters f ,r,r0 and r1 are rational functions. There are several algorithms solving special cases of the transformation mentioned above. If one just allows exp-products and gauge transformations, the prob- vii
Page 1 and 2: Solving Differential Equations in T
Page 3 and 4: Contents List of Notations v Introd
Page 5: List of Notations This is a list of
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 and 18: 1.3. HYPERGEOMETRIC SERIES 17 Remar
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
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4 Conclusion We developed an algori
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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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