Index 975 power spectrum estimation by maximum associated Legendre polynomial 252f, entropy method 572ff. 773 properties of 497f. autocorrelation of 498 sampling theorem 501,550,552,606f bandwidth limited 501 scalings of 497 Bessel178.210.230f.,240f significance of a peak in 577f. beta 215f sine transform 514ff.,859 branch cuts of 209f. symmetries of 497 chi-square probability 221,806 uneven sampling,fast algorithm 581f. complex 208 unevenly sampled data 575ff..581f. confluent hypergeometric 210,246 and wavelets 599f. convolution of 498 Wiener-Khinchin theorem 498,566,574 correlation of 498 see also Fast Fourier transform (FFT); Coulomb wave 210,240 Spectral density cumulative binomial probability 226,229 Fractal region 367f. cumulative Poisson 216 Fractional step methods 856f. Dawson's integral 259f.,606 Fredholm alternative 789 declaration 17 Fredholm equations 788f definition 17 eigenvalue problems 789,794 digamma 222 elliptic integrals 261ff,915 error estimate in solution 793 first kind 788 eror220f,255,259.607.636.641 evaluation 165ff Fredholm alternative 789 evaluation by path integration 208ff.,271 homogeneous,second kind 793f. homogeneous vs.inhomogeneous 789 exponential integral 178,222ff.,257 extemal 25 ill-conditioned 789 F-distribution probability 226,229 infinite range 797E Fresnel integral 255ff. inverse problems 789,804ff gamma 213f. kernel 788f. hypergeometric 208ff.,271ff. nonlinear 790 incomplete beta 226ff.,616 Nystrom method 791ff.,797f. incomplete gamma 216ff.,621,660.663f. product Nystrom method 797 inverse hyperbolic 184,262 second kind 789,791f. inverse trigonometric 262 with singularities 797 Jacobian elliptic 261,269f with singularities,worked example 801 Kolmogorov-Smimov probability 624f., subtraction of singularity 798 646f symmetric kernel 794 Legendre polynomial 178,252f.,680 see also Inverse problems logarithm 262 Freeing of storage 19,21f.,940ff modified Bessel 236f. free_matrix()utility 946 modified Bessel,fractional order 246ff. free_vector()utility 946 path integration to evaluate 208ff. Frequency domain 496 pathological 105f,350f. Frequency spectrum see Fast Fourier transform Poisson cumulant 221 (FFT) prototypes 16f.,25,930 Frequentist,contrasted with Bayesian 819 representations of 496 Fresnel integrals 255ff. routine for plotting a 349f. asymptotic form 255 sine and cosine integrals 255,257ff. continued fraction 255 sn,dn,cn 269 routine for 256f. spherical Bessel 240 series 255 spherical harmonics 252f. Friday the Thirteenth 13f. spheroidal harmonic 772ff,779f.,781 FTCS(forward time centered space)836ff., Student's probability 226,228 847f,852 Weber 210 stability of 836ff..847ff..864 Functional iteration,for implicit equations 748 Full approximation storage (FAS)algorithm 882f FWHM (full width at half maximum)555 Full moon 13f Full multigrid method (FMG)872,877f. Gamma deviate 290ff. Full Newton methods,nonlinear least squares Gamma function 213ff 688 incomplete see Incomplete gamma func- Full pivoting 38 tion Full weighting 876 Gauss-Chebyshev integration 147,151,518f. Function Gauss-Hermite integration 151,798 Aiy210,240,250 abscissas and weights 153 approximation 105f.,190ff. normalization 153Index 975 power spectrum estimation by maximum entropy method 572ff. properties of 497f. sampling theorem 501, 550, 552, 606f. scalings of 497 significance of a peak in 577f. sine transform 514ff., 859 symmetries of 497 uneven sampling, fast algorithm 581f. unevenly sampled data 575ff., 581f. and wavelets 599f. Wiener-Khinchin theorem 498, 566, 574 see also Fast Fourier transform (FFT); Spectral density Fractal region 367f. Fractional step methods 856f. Fredholm alternative 789 Fredholm equations 788f. eigenvalue problems 789, 794 error estimate in solution 793 first kind 788 Fredholm alternative 789 homogeneous, second kind 793f. homogeneous vs. inhomogeneous 789 ill-conditioned 789 infinite range 797f. inverse problems 789, 804ff. kernel 788f. nonlinear 790 Nystrom method 791ff., 797f. product Nystrom method 797 second kind 789, 791f. with singularities 797 with singularities, worked example 801 subtraction of singularity 798 symmetric kernel 794 see also Inverse problems Freeing of storage 19, 21f., 940ff. free_matrix() utility 946 free_vector() utility 946 Frequency domain 496 Frequency spectrum see Fast Fourier transform (FFT) Frequentist, contrasted with Bayesian 819 Fresnel integrals 255ff. asymptotic form 255 continued fraction 255 routine for 256f. series 255 Friday the Thirteenth 13f. FTCS (forward time centered space) 836ff., 847ff., 852 stability of 836ff., 847ff., 864 Full approximation storage (FAS) algorithm 882ff. Full moon 13f. Full multigrid method (FMG) 872, 877f. Full Newton methods, nonlinear least squares 688 Full pivoting 38 Full weighting 876 Function Airy 210, 240, 250 approximation 105f., 190ff. associated Legendre polynomial 252f., 773 autocorrelation of 498 bandwidth limited 501 Bessel 178, 210, 230ff., 240ff. beta 215f. branch cuts of 209f. chi-square probability 221, 806 complex 208 confluent hypergeometric 210, 246 convolution of 498 correlation of 498 Coulomb wave 210, 240 cumulative binomial probability 226, 229 cumulative Poisson 216 Dawson’s integral 259f., 606 declaration 17 definition 17 digamma 222 elliptic integrals 261ff., 915 error 220f., 255, 259, 607, 636, 641 evaluation 165ff. evaluation by path integration 208ff., 271 exponential integral 178, 222ff., 257 external 25 F-distribution probability 226, 229 Fresnel integral 255ff. gamma 213f. hypergeometric 208ff., 271ff. incomplete beta 226ff., 616 incomplete gamma 216ff., 621, 660, 663f. inverse hyperbolic 184, 262 inverse trigonometric 262 Jacobian elliptic 261, 269f. Kolmogorov-Smirnov probability 624f., 646f. Legendre polynomial 178, 252f., 680 logarithm 262 modified Bessel 236ff. modified Bessel, fractional order 246ff. path integration to evaluate 208ff. pathological 105f., 350f. Poisson cumulant 221 prototypes 16f., 25, 930 representations of 496 routine for plotting a 349f. sine and cosine integrals 255, 257ff. sn, dn, cn 269 spherical Bessel 240 spherical harmonics 252f. spheroidal harmonic 772ff., 779f., 781 Student’s probability 226, 228 Weber 210 Functional iteration, for implicit equations 748 FWHM (full width at half maximum) 555 Gamma deviate 290ff. Gamma function 213ff. incomplete see Incomplete gamma func￾tion Gauss-Chebyshev integration 147, 151, 518f. Gauss-Hermite integration 151, 798 abscissas and weights 153 normalization 153