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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
±
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
→
∞
§13.2(iv)
as
z
→
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
F
2
3
hypergeometric functions of matrix argument
§35.8(iii)
for confluent hypergeometric functions
§13.2(vii)