Absolutely integrable 2 31-4 46-56
Absolutely integrable in every finite interval 48-56
Arbitrary constant in definitions of particular generalised functions 39-43
Asymptotic estimation 10
Asymptotic estimation of Fourier coefficients 75
Asymptotic estimation of Fourier transforms 46-57
Asymptotic expansion 50 54-7
Bessel functions 50-1 53-7
boundary conditions 4-5 9
Cauchy principal value 14 37-8
Classical theory, of Fourier series 7 60 66
Classical theory, of Fourier transforms 8-9 34
Constant generalised function 17
Constant, arbitrary, in definitions of particular generalised functions 39-43
Contour integration 2 33 44
Convergence of Fourier series 6-7 58-60
Delta function 1 9-13 33 25-9 31 36-41 48
Delta function, definition of 17
Delta function, derivatives of 11-13 19 36-7 29 32 36 39 43-5 52 61 68
Differential equation 1 3-10
Differentiation, of generalised function 18 24-5
Differentiation, of series term by term 3-4 7 29 59 68-9
Differentiation, with respect to a parameter 39 34-5
Dipole 13
Dirac, P. A. M., i 9-10
Elliptic function 70
Equivalent regular sequences 17
Error term 46 53
Error term, precise order of magnitude of 55 7 74-5
Even function 5-7 10 26-7 31 38 41
Factorial function 33-6 51 53-4
Fairly good function 15 18-20 38-9 65 72
Fourier coefficients 3-7 60-75
Fourier coefficients, asymptotic estimation of 71-5
Fourier cosine integral 10 56-7
Fourier cosine series 5 7 68-71 75
Fourier inversion theorem, for generalised functions 20
Fourier inversion theorem, for good functions 16
Fourier inversion theorem, for particular generalised functions 34 40-4
Fourier series 1 3-7 58-75
Fourier series, classical theory of 7 60 66
Fourier series, convergence of 6-7 58 60
Fourier series, differentiation of 3-4 7 29 67-9
Fourier series, existence of, for any periodic generalised function 62-6
Fourier series, half-range, quarter-range 4-7 74-5
Fourier series, summability of 14 68
Fourier series, uniqueness of representation by 4 7 58 60 68
Fourier sine integral 9 56-7
Fourier sine series 5 7 68 74-5
Fourier transform (f.t.) 8 10
Fourier transform (f.t.) of derivative of generalised function 30 33
Fourier transform (f.t.) of generalised function 18-21 34 27
Fourier transform (f.t.) of good function 15-16
Fourier transform (f.t.) of ordinary function 24
Fourier transform (f.t.) of particular generalised functions 30-45
Fourier transform (f.t.) of periodic function 58-68
Fourier transform (f.t.) of rational function 42-5
Fourier transform (f.t.), asymptotic estimation of 1
Fourier transform (f.t.), classical theory of 8-9 24
Fourier transform (f.t.), definitions of 8 15 18
Fourier transform (f.t.), limit of 28-9 36 40-1
Fourier transform (f.t.), summary of results on 42-5
Fourier transform (f.t.), table of 43
frequency 8 49
Generalised function 1 10-14
Generalised function derivative of with respect to a parameter 29 34-5
Generalised function, constant 17
Generalised function, definition of 17
Generalised function, derivative of, J 8 34-5
Generalised function, equal to ordinary function in an interval 35-6
Generalised function, general properties 16-29
Generalised function, limit of 27-9
Generalised function, limit of derivative of 28-9 33 39
Generalised function, limit of F.T. of 28-9 36 40-1
Generalised function, ordinary function as 21-4
Generalised function, particular 30-45 53-7
Generalised function, periodic 60-75
| Generalised function, series of 29 58-75
Generalised function, well behaved at infinity 49-57 73
Good function 13-16
Good function, definition of 15
Good function, fairly 15 18-20 28-9 65 72
Good function, sequence of 10-14 16-18 21-3 65-6 70
Hadamard finite part 14 31-2 34
Half-range, Fourier integral 9-10 56-7
Half-range, Fourier series 4-7 74-5
Hardy, G. H. 7
Heaviside unit function H(x) 30-45 55-8 69
Historical account of subject 13
Improper integral 31-2 37-8
Indefinite integral 14 38
Integral powers 35-40
Integral powers, multiplied by logarithms 40-2
Integration byparts 11 15 19 31-2 37
Interchange of order, of limit processes and differentiation 8 9 35
Interchange of order, of limit processes and Fourier transformation 38-9 36 40-1
Interchange of order, of summation and/or integration 2 16 22-4 59 64-6
Jacobi's transformation 70
Laplace's equation 3-4
Laurent's theorem 4
Lebesgue theory of integration 2
Limit, of derivatives of generalised functions 28-9 33 39
Limit, of f.t.'s of generalised functions 28-9 36 40-1
Limit, of generalised functions 27-9
Logarithmic singularities 34-5 40-3 52 56-7 75
Mikusfnski, J. G. 13
Non-integral powers 30-4
Odd function 5-7 9 26-7 31 36
Ordinary function, as generalised function 21-4
Ordinary function, derivative of 23-5
Ordinary function, equal to generalised function in an interval 25-6
oscillations 3 8 49
Parseval's theorem 16 19 69
Partial fractions 42-5
Particular generalised functions 30 45 53-7
Particular generalised functions, arbitrary constant in definition of 39-43
Periodic function 3-7
Periodic function, equal to given function in an interval 5-6 8 69
Periodic function, generalised 60 75
Point charge 12
Poisson summation formula 67-71
Polar coordinates 3-4
Powers, integral 35-40
Powers, non-integral 30-4
Psi function (a) 35 39-43
Quarter-range Fourier series 4-6
Rational function, F.T. of 42-5
Regular sequence 16-18 21-3 65-6
Remainder 46 52
Riemann - Lebesgue lemma 46-51
Rogosinski, W. W. 7
Row of delta functions, i 60 67-9
Schwartz, L., I 14
Sequence of good functions 10-14 16-18 21-3 65-6
Series of generalised functions 29 58-75
Sgn function 23 28 30-45 47 49 52 54 56-7 72 74
Singularity 42 51-7 72-5
Singularity, defined 51 72
Singularity, worst 55 57 75
Smudge function 22
Step function 58-9
String, plucked 5-6
Summability of Fourier series 14 68
Summary of Fourier transform results 42-45
Temple. G. 1 8 13-14
Test function 13-14
Theta function 70
Titchmarsh, E. C. 8
Unitary function 61-7 71-3
Watson, G. N. 51
wave number 8
Waveform 3
Well behaved at infinity 49-57 73
'Worst' singularity 55 57 75
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