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
§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
-
zeros
- asymptotic approximations for large zeros §10.70
- computation §10.74(vi)
- tables §10.75(xi)
- Kelvin’s ship-wave pattern §36.13
-
kernel equations
- Heun’s equation ¶ ‣ §31.10(i), ¶ ‣ §31.10(ii)
-
kernel functions
- Heun’s equation ¶ ‣ §31.10(i), ¶ ‣ §31.10(ii)
-
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)
-
modified Bessel functions §10.43(v)
-
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)
-
applications
-
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)
- analytical properties ¶ ‣ §13.2(i)
- analytic continuation §13.2(ii)
-
applications
- physical §13.28
- approximations §13.31
-
asymptotic approximations for large parameters
-
large
§13.8(iii), §13.8(iii)
-
large
§13.8(i), §13.8(ii)
- uniform §13.8(ii), §13.8(iii)
-
large
-
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
§13.2(iii)
-
as
§13.2(iv)
-
as
- 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
-
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 confluent hypergeometric functions ¶ ‣ §13.2(vii)
-
for
hypergeometric functions of matrix argument ¶ ‣ §35.8(iii)