a + i b form 113
Abel's formula 547
Absolute convergence 320
adjoint of a differential operator 550
adjoint of operators 787
Analytic 219
Analytic continuation 263
analytic continuation Fourier integrals 933
Analytic functions 1367
Anti-derivative 285
Argand diagram 113
Argument of a complex number 114
argument theorem 303
Asymptotic expansions 747
asymptotic integration by parts 754
asymptotic relations 747
autonomous D.E. 593
Autonomous differential equations 593
average value theorem 302
behavior of Fourier coefficients 811
Bernoulli equations 589
Bessel functions 979
Bessel generating function 983
Bessel generating function of the first kind 983
Bessel generating function second kind 992
Bessel's equation 979
Bessel's inequality 774
Bijective function 5
Bilinear concomitant 551
Binomial coefficients 1409
Binomial formula 1409
Boundary value problems 664
Branch point 165
branch point singularity 229
branch principal 6
Branches 6
Calculus of variations 1261
canonical forms constant coefficient equation 609
canonical forms of differential equations 609
Cardinality of a set 4
Cartesian form 113
Cartesian product of sets 4
Cauchy convergence 319
Cauchy principal value 384 932
Cauchy's inequality 301
Cauchy—Riemann equations 223 1367
clockwise 148
Closed curve 147
Closed interval 3
closure relation and Fourier transform 934
closure relation discrete sets of functions 775
Codomain 4
Comparison Test 321
comparison test series 321
completeness of sets of functions 774
completeness of sets of vectors 24
Complex conjugate 112 113
complex derivative 219
complex infinity 149
Complex line integral 280
complex magnitude 113
complex modulus 113
Complex number 112
complex numbers 111
complex numbers arithmetic 118
Complex plane 113
components of vector 18
computer games 1413
connected region 147
constant coefficient differential equations 557
Continuity 36
Continuity uniform 37
continuous curve 147
Continuous functions 36 325 327
continuous piecewise 37
contour 147
Contour integral 280
convergence comparison test 321
Convergence in the mean 774
convergence integral test 322
convergence of integrals 885
Convergence of sequences 319
Convergence of series 319
convergence, Raabe's test 325
convergence, ratio test 323
convergence, root test 324
convolution theorem and Fourier transform 936
Convolution theorem for Laplace transforms 898
convolution theorem of Laplace transforms 898
Convolutions 898
counter-clockwise 148
Curve 147
Definite integral 76
definition of power series 328
degree of a differential equation 467
DEL 95
derivative complex 219
derivative of determinant 544
difference constant coefficient equations 699
difference exact equations 696
difference first order homogeneous 697
difference first order inhomogeneous 698
Difference of sets 4
Differential calculus 33
differential equations constant coefficient 557
differential equations degree 467
differential equations equidimensional-in-x 595
differential equations equidimensional-in-y 596
differential equations first order 467 476
differential equations order 467
differentiation of power series 332
Dirac delta function 625 775
Directional derivative 95
discontinuous functions 36 802
discrete derivative 695
discrete integral 695
Disjoint sets 4
Domain 4
Economics 1413
Eigenfunctions 799
eigenvalue problems 799
Eigenvalues 799
Elements of a set 3
Empty set 3
Entire 219
equidimensional differential equations 563
equidimensional-in-x D.E. 595
equidimensional-in-y D.E. 596
Euler differential equations 563
Euler's formula 116
Euler's formula of Gamma function 967
Euler's notation i 112
Euler's theorem 473
Euler—Mascheroni constant 970
exact differential equations 469 566
exact equations 469
exchanging dep. and indep. var. 592
Extended complex plane 149
extremum modulus theorem 303
Fibonacci sequence 703
first order differential equations 484
fluid flow ideal 232
formally self-adjoint operators 788
Fourier coefficients 771 802
Fourier convolution theorem 936
Fourier cosine series 808
Fourier cosine transform 940
Fourier cosine transform of derivatives 942
Fourier series 799
Fourier series and Fourier transform 927
| Fourier series uniform convergence 814
Fourier sine series 809
Fourier sine transform 941
Fourier sine transform of derivatives 942
Fourier transform alternate definitions 929
Fourier transform closure relation 934
Fourier transform convolution theorem 936
Fourier transform of a derivative 935
Fourier transform shift property 938
Fredholm alternative theorem 664
Fredholm equations 614
Frobenius series first order differential equation 487
function elements 263
Functional equation 236
fundamental set of solutions of a differential equation 549
Fundamental theorem of algebra 302
Fundamental theorem of calculus 78
Gambler's ruin problem 695 700
Gamma function 967
Gamma function difference equation 967
Gauss' formula of Gamma function 969
Gauss' test 325
Gauss' test convergence 325
Gauss' test series 325
generating function for Bessel functions 983
Geometric series 320
Gibb's phenomenon 817
Gradient 95
Gramm—Schmidt orthogonalization 766
Greatest integer function 5
Green's formula 551 787
Hankel's formula of Gamma function 968
Harmonic conjugate 226
Harmonic series 321 342
Heaviside function 479 625
Holomorphic 219
homogeneous coefficient differential equations 473
homogeneous coefficient equations 473
homogeneous differential equations 467
homogeneous functions 473
homogeneous solution 477
homogeneous solutions of differential equations 542
i Euler's notation 112
ideal fluid flow 232
identity map 4
ill-posed problems 482
image of a mapping 4
Imaginary number 112
Imaginary part 112
Improper integrals 81
Indefinite integral 73 285
Indicial equation 718
infinity complex 149
infinity first order differential equation 490
inhomogeneous differential equations 467
Initial conditions 478
Injective function 5
inner product of functions 769
integral bound maximum modulus 282
Integral calculus 73
Integral equations 614
integral equations of boundary value problems 614
integral equations of initial value problems 614
integral test series 322
Integrals improper 81
Integrating factor 477
integration of power series 332
Intermediate Value Theorem 37
Intersection of sets 4
Inverse function 5
Inverse image 4
inverse of function 5
irregular first order differential equations 489
irregular singular points 726
j electrical engineering 112
Jordan curve 148
Jordan's lemma 1367
Kramer's rule 1403
Kronecker delta function 24
l'Hospital's rule 49
Lagrange's identity 551 568 787
Laplace transform inverse 890
Laplace transform pairs 892
Laplace transforms 889
Laplace transforms of derivatives 898
Laurent expansions 379 1367
Laurent series 337 1368
Laurent series first order differential equation 486
leading order behavior for differential equations 749
Least integer function 5
least squares fit Fourier series 803
Legendre polynomials 767
limit left and right 34
limits of functions 33
Line integral 280
Linear differential equations 467 547
Linear differential operator 542
Linear space 763
Liouville's theorem 301
Magnitude 113
maximum modulus integral bound 282
Maximum modulus theorem 303
Mellin inversion formula 890
minimum modulus theorem 303
Modulus 113
multi-valued function 5
multiply-connected region 147
nabla 95
Natural boundary 263
negative direction 148
Newton's binomial formula 1409
norm of functions 769
normal form of differential equations 611
Null vector 18
One-to-one mapping 5
Open interval 3
Opportunity cost 1413
optimal asymptotic approximations 757
Order of a differential equation 467
Order of a set 4
Ordinary differential equations 467
ordinary points first order differential equations 484
ordinary points of linear differential equations 707
orthogonal series 771
orthogonality weighting functions 770
Orthonormal 769
Parseval's equality 805
Parseval's theorem for Fourier transform 938
Partial derivative 94
Particular solution 477
particular solution of an ODE 635
particular solutions of differential equations 542
periodic extension 802
Piecewise continuous 37
Piecewise smooth curve 147
Point at infinity 149
point differential equations 726
Polar form 115
Positive direction 148
potential flow 232
principal argument 114
Principal branch 6
principal root 122
Principal value 384 932
properties of regular Sturm—Liouville problems 860
pure imaginary number 112
Raabe's test 325
Raabe's test series 325
radius of convergence power series 328
range of a mapping 4
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