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| Small Ch.G. — Functional Equations and how to Solve Them |
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| Предметный указатель |
Abelian group 88
Abel’s equation 15 63 64 67 76
Abel’s equation, generalization 67
Acceleration 3
Aczel, J. 3 123
Alembert, J.d’ 9
Alembert, J.d’, also known as Daremberg 9
Alembert, J.d’, also known as Rond, J.le 9
Alembert, J.d’, Daremberg changed to d’Alembert 9
Alembert, J.d’, d’Alembert’s equation 10 17 18 45
Alembert, J.d’, law school 9
Alembert, J.d’, parents and background 9
Alembert, J.d’, placed in orphanage 9
Alembert, J.d’, solutions to d’Alembert’s equation 10
Alembert, J.d’, supported by father financially 9
Algebra 1
Analysis 6 7
Area 6
Aristotle 2
Arithmetic f.e. 82
Arithmetic function 82
Arithmetic mean 51
Average 37
Axiom of Choice 93 96
Axiom of Choice, nonconstructive existence theorem 96
Babbage, C. 10 14 15 19 55 61
Babbage, C., analytical engine 11
Babbage, C., cowcatcher 11
Babbage, C., difference engine 11
Babbage, C., essay on calculus of functions 12
Babbage, C., flat rate postage 11
Babbage, C., skeleton keys 11
Bastille 6
Big Omega 84
Binet’s formula 20
Binomial coefficients 7
Binomial theorem 7 35 38
Black Death 1 2
Bonaparte, N. 7
Bottcher’s equation 63 66
British mathematics 11
Calculus 6—8 17 18
Calculus, differential equations 17 18
Calculus, integral equations 17
Calculus, Leibniz’s methodology 11
Calculus, Newtonian methodology 11
Cambridge Analytical Society 11
Cambridge Analytical Society, renamed Cambridge Philosophical Society 11
Canadian Mathematical Society 16
Carr, G.S. 22
Cauchy, A.-L. 6—9 18 31 93
Cauchy, A.-L., Cauchy’s equation 8—10 18 31 35
Cauchy, A.-L., Cauchy’s exponential equation 38
Cauchy, A.-L., early years 6
Cauchy, A.-L., family 7
Cauchy, A.-L., founder of analysis 7
Cauchy, A.-L., full solution to d’Alembert’s equation 10
Cauchy, A.-L., problems with republicans and Napoleonic movements 7
Cauchy, A.-L., royalist sympathies 7
Cauchy, A.-L., starting point for theory of f.e. 6
Cauchy’s equation 7 24 31 35 37—40 43 95
Cauchy’s equation for functions bounded below 34
Cauchy’s equation for monotone function 35
Cauchy’s equation on a circle 50
Cauchy’s equation, solution on rationals 33
Cauchy’s equation, solution on reals 34
Cauchy’s equation, tame solutions 96
Cauchy’s equation, two variables 43
Cauchy’s equation, two-variable solution 44
Cauchy’s equation, wild solutions 96
Cauchy’s exponential equation 8 38 39 42
Cauchy’s inequality 42 50
Challenges and puzzles 16
Circular cosine 10
Clagett translation 2
Classification of problems 18
Closure, under multiplication 88
Closure, under reciprocation 88
College of Navarre, College of 2
Combinations of forces 9
Combinatorial arguments 8
Commutative group 88
Commutativity 41
Commutativity equation 63 67 76
Commutativity equation, solutions for 67
Competition 1 16—19 22 84
Competition, 1995 Romanian Winter 80
Competition, IMO 17
Competition, IMO 1972 19
Competition, IMO 1997 shortlist 56 71 76
Competition, IMO 1998 84
Competition, IMO 2001 88
Competition, IMO 2001 jury 88
Competition, IMO 2001 shortlisted problem 87
Competition, IMO gold medal problem 84
Competition, IMO medals 84
Competition, problems 17
Competition, Putnam 17 18
Competition, Putnam 1947 18
Competition, Putnam 1955 18
Competition, Putnam 1956 18
Competition, Putnam 1959 19
Competition, Putnam 1963 19
Competition, Putnam 1966 22
Competition, Putnam 1971 62
Competition, Putnam 1972 20
Competition, Putnam 1979 70
Competition, Putnam 1988 21
Competition, Putnam 1996 59
Completely multiplicative function 83
Composition of functions 69
Computing 11
Conic sections 4 6
Conjugacy equation 15 62 64 67
Conjugacy equation, finding solutions 64
Constant of integration 17
Continued fractions 22
Cosine function 10 46 48
Cosine function, double angle formula 46
Cosine function, summation formula 48
Cosine, circular 10 48
Cosine, hyperbolic 10
Crux Mathematicorum 16
Cyclic function 61
Degree of polynomial 18 79
Dense set 47
Descartes, R. 2
Descartes, R., graphs of functions 2
Dhombres, J. 3 123
Differential equations versus f.e. 19
Differentiation 8 9
Difform intensities 2
Direct and inverse calculation 12
Direct and inverse calculation, inverse more difficult 12
Directoire 7
Distance 3
Distance, signed 3
Divergent series 22
Dubikajtis’ theorem 112
d’Alembert’s equation 45 49
Elementary methods without calculus 18
Enlightenment 2
Equation in two variables 10
Equivalence class 58
Equivalence relation 58
Equivalence relation, reflexivity 58
Equivalence relation, symmetry 58
Equivalence relation, transitivity 58
Euler’s equation 39 44
Euler’s equation, generalized 45
| Euler’s equation, solution 44
Euler’s equation, solution to generalized 50
Europe 1
Exhaustion, method of 6
Extreme Value Theorem 34
Fibonacci numbers 20
Fixed point under iteration 59
Fourteenth century 1
Fractional part 58
France 7
French Revolution 6
Function of two variables 43
Function, analytic 81
Function, antisymmetric 14
Function, arithmetic 82
Function, bounded on interval 7 35 45
Function, completely multiplicative 83 84
Function, complex-valued 77
Function, constant 4 9 58
Function, continuous 7—9 18 25—28 31 33—35 37—40 43—45 49—51 58 59 69 70 75 76 78 101
Function, decreasing 35 51 69 70 89
Function, differentiable 9
Function, even 47
Function, homogeneous 44
Function, homogeneous of degree k 44
Function, increasing 3 35—37 51 69 70 89 99 107 116
Function, involution 13—15 61
Function, linear 1 3 6 7 9
Function, monotone 35 75
Function, nondecreasing 89 120
Function, nowhere continuous 9
Function, nowhere differentiable 9
Function, one-to-one 14 15
Function, periodic 77 80 92
Function, positive 39
Function, quadratic 2
Function, restricted domain 18
Function, strictly positive 21
Function, symmetric 13 52
Functional equation, commutativity equation 63 76
Functional equation, conjugacy equation 62 64
Functional equation, existence of solution 26
Functional equation, families of 57
Functional equation, miscellaneous methods 79
Functional equation, nonlinear 23
Functional equation, one variable 55
Functional equation, polynomial 18 79 80
Functional equation, simultaneous 24
Functional equation, single variable 25
Functional equation, symmetries in 91
Functional equation, too general 42
Functional equation, two or more variables 25 31
Functional equation, uniqueness of solution 26
Functional equations versus differential equations 19
Gamma function 75 107
Geometric mean 51
Golden ratio 20
Golden ratio as limiting ratio of Fibonacci sequence 20
Gottingen University 93
Group 88
Group, commutative 88
Group-theoretic methods 87
Hamel bases 35 93 95
Hamel bases, uncountable 95
Hamel bases, uniqueness of representation 94
Hamel, G. 35 93
Hamel, G., 1905 paper of Hamel bases 93
Hardy — Ramanujan number 73
Hardy, G.H. 73
Harmonic mean 51
Herschel, J. 11
Herschel, W. 11
Heuristics for solving f.e. 91
Heytesbury, William of 2
High school mathematics 16
High school students 16
Hilbert, D. 93
Hilbert’s fourth problem 93
Hints to problems 97
Hints to problems, Chapter 1 97
Hints to problems, Chapter 2 102
Hints to problems, Chapter 3 107
Hints to problems, Chapter 4 113
history 1 17 123
Hyperbola 4—6
Hyperbolic cosine 10 46 47
Hyperbolic cosine, summation formula 48
Hyperbolic sine 48
Hyperbolic tangent 48
Identity 88
Identity function 14
Indeterminate expression 18
Induction 24 31 46 47 83 85
Ingenuity in solutions 91
Integration 4
Inverse problem 12
Inverse problem, f.e.as 12
Italian Renaissance 16
Iteration 58
Iterative roots 14 15 19 55 61 69
Iterative square root 14 20
Jensen’s equation 37 38
Jensen’s equation, solution 38
Journal of the Indian mathematical society 23
Klein, F. 93
Koenigs algorithm 64 76 81
Legendre symbol 84
Levy algorithm 66
LIMIT 25 59 64 66
Linear algebra 62
Linear functional equation 38
Linear functional equation, solution 38 51
Linear functions 91
Linearization 55 92
Liouville function 84
Lisieux 2
Logarithms 4—6 8 39 91
Mathematical Association of America 16
Mathematical Mayhem 16
Mathematics Magazine 16
Mean speed theorem 2
Mechanics 2
Medieval world 2
Merton College 2
Merton theorem 2
Multiplicative group 88
Multiplicative identity 88
Multiplicative inverse 88
Natural philosophy 2
Natural sciences 2
Nested radical 21—23 72
Nested radicals 72
Newton, I. 7
Newton, I., derivation of binomial theorem 7
Nonconstructive existence 96
Nonlinear equation 23
Nth order f.e. 13
Number theory 84
Numbers, complex 77 79
Numbers, dyadic rationals 47
Numbers, Hardy — Ramanujan number 73
Numbers, perfect squares 23
Numbers, positive integers 82
Numbers, primes 84 85
Numbers, rational 32 33 50
Numbers, real 79
Orbit 58
Oresme 2
Oresme, N. 1—4 6
Oresme, N., defined functional relationship 2
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