Index K

Kadomtsev–Petviashvili equation
- Riemann theta functions §21.9
Kapteyn’s inequality
- Bessel functions §10.14
KdV equation §22.19(iii), §23.21(ii), §32.13(i), §9.16
Kelvin functions Ch.10
- applications §10.73(iii)
- approximations §10.76(iii)
- asymptotic expansions for large argument §10.67(i)—§10.67(i)
  - cross-products and sums of squares §10.67(ii)
  - exponentially-small contributions §10.67(i)
- asymptotic expansions for large order, see uniform asymptotic expansions for large order.
- computation Ch.10—§10.74(v)
- cross-products §10.63(ii)
- definitions §10.61(i)
- derivatives §10.63(i)
  - with respect to order §10.64
- differential equations §10.61(ii)
- expansions in series of Bessel functions §10.66
- graphs §10.62
- integral representations §10.64
- integrals
  - compendia §10.71(iii)
  - definite §10.71(ii)—§10.71(iii)
  - indefinite §10.71(i)
  - Laplace transforms §10.71(iii)
- modulus and phase functions
  - asymptotic expansions for large argument §10.68(iii)
  - definitions §10.68(i)
  - properties §10.68(ii)—§10.68(ii)
- notation §10.1
- orders $\pm\frac{1}{2}$ §10.61(v)
- power series §10.65—§10.65(iv)
  - compendia §10.65(iv)
  - cross-products and sums of squares §10.65(iii)
- recurrence relations §10.63(i)—§10.63(ii)
- reflection formulas for arguments and orders §10.61(iii), §10.61(iv)
- uniform asymptotic expansions for large order §10.69—§10.69
  - double asymptotic property §10.69
  - exponentially-small contributions §10.69
- zeros
  - asymptotic approximations for large zeros §10.70
  - computation §10.74(vi)
  - tables §10.75(xi)
Kelvin’s ship-wave pattern §36.13—§36.13
kernel equations
- Heun’s equation §31.10(i), §31.10(ii)
kernel functions
- Heun’s equation §31.10(i), §31.10(ii)
kernel polynomials §18.2(v)
Klein–Gordon equation
- Coulomb functions §33.22(iv)
Klein’s complete invariant, see modular functions.
Kontorovich–Lebedev transform
- modified Bessel functions §10.43(v)
  - computation §10.74(vii)
Korteweg–de Vries equation
- Airy functions §9.16
- Jacobian elliptic functions §22.19(iii)
- Lamé polynomials §29.19(ii)
- Painlevé transcendents §32.13(i)
- Riemann theta functions §21.9
- Weierstrass elliptic functions §23.21(ii)
Kovacic’s algorithm §31.14(ii), §31.8
KP equation, see Kadomtsev–Petviashvili equation.
Krattenthaler’s formula for determinants §1.3(ii)
Krawtchouk polynomials, see also Hahn class orthogonal polynomials.
- applications
  - coding theory §18.38(iii)
- relation to hypergeometric function §15.9(i)
Kummer congruences
- Bernoulli and Euler numbers §24.10(ii)
Kummer functions Ch.13, see also confluent hypergeometric functions.
- addition theorems §13.13(i)—§13.13(ii)
- analytic continuation §13.2(ii)
- analytical properties §13.2(i)
- applications
  - physical §13.28
- approximations §13.31
- asymptotic approximations for large parameters
  - large $a$ §13.8(iii)—§13.8(iii)
  - large $a$ and $b$ §13.8(iv)
  - large $b$ §13.8(i), §13.8(ii)
  - uniform §13.8(ii)—§13.8(iii)
- asymptotic expansions for large argument §13.7—§13.7(iii)
  - error bounds §13.7(ii)
  - exponentially-improved §13.7(iii)
  - hyperasymptotic §13.7(iii)
- Chebyshev-series expansions §13.31(i)
- computation Ch.13—§13.31(iii)
- connection formulas §13.2(vii)
- continued fractions §13.5
- definitions §13.2
- derivatives §13.3(ii)—§13.3(ii)
- differential equation, see Kummer’s equation
- integer parameters §13.2(i)—§13.2(i)
- integral representations
  - along the real line §13.4(i)
  - contour integrals §13.4(ii)—§13.4(iii)
  - Mellin–Barnes type §13.4(iii)
- integrals
  - along the real line §13.4(i)
  - compendia §13.10(vi)
  - Fourier transforms §13.10(iv)
  - Hankel transforms §13.10(v)—§13.10(v)
  - indefinite §13.10(i)
  - Laplace transforms §13.10(ii)
  - Mellin transforms §13.10(iii)
- interrelations §13.2(i), §13.2(vii)
- Kummer’s transformations §13.2(vii)
- limiting forms
  - as $z\rightarrow\infty$ §13.2(iv)
  - as $z\rightarrow 0$ §13.2(iii)
- Maclaurin series §13.2(i)
- multiplication theorems §13.13(iii)
- notation §13.1
- polynomial cases §13.2(i), §13.2(i)
- principal branches (or values) §13.2(i)
- products §13.12
- recurrence relations §13.3(i)
- relations to other functions
  - Airy functions §13.6(iii)
  - elementary functions §13.6(i)
  - error functions §13.6(ii)
  - generalized hypergeometric functions §13.6(vi)
  - incomplete gamma functions §13.6(ii)
  - modified Bessel functions §13.6(iii)
  - orthogonal polynomials §13.6(v)
  - parabolic cylinder functions §13.6(iv)
  - Whittaker functions §13.14(i)
- series expansions
  - addition theorems §13.13(i)—§13.13(ii)
  - in modified Bessel functions §13.11
  - Maclaurin §13.2(i)
  - multiplication theorems §13.13(iii)
- tables §13.30
- Wronskians §13.2(vi)
- zeros
  - asymptotic approximations §13.9(i), §13.9(i)
  - distribution §13.9(i)
  - inequalities §13.9(i)
  - number of §13.9(i), §13.9(ii)

Kummer’s equation §13.2(i)
- equivalent form §13.3(i)
- fundamental solutions §13.2(v)—§13.2(v)
- numerically satisfactory solutions §13.2(v)—§13.2(v)
- relation to hypergeometric differential equation §13.2(i)
- relation to Whittaker’s equation §13.14(i)
- standard solutions §13.2(i)
Kummer’s transformations
- for ${{}_{3}F_{2}}$ hypergeometric functions of matrix argument §35.8(iii)
- for confluent hypergeometric functions §13.2(vii)

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