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1: 12.8 Recurrence Relations and Derivatives
12.8.5 z V ( a , z ) V ( a + 1 , z ) + ( a 1 2 ) V ( a 1 , z ) = 0 ,
12.8.6 V ( a , z ) 1 2 z V ( a , z ) ( a 1 2 ) V ( a 1 , z ) = 0 ,
12.8.7 V ( a , z ) + 1 2 z V ( a , z ) V ( a + 1 , z ) = 0 ,
12.8.8 2 V ( a , z ) V ( a + 1 , z ) ( a 1 2 ) V ( a 1 , z ) = 0 .
12.8.11 d m d z m ( e 1 4 z 2 V ( a , z ) ) = e 1 4 z 2 V ( a + m , z ) ,
2: 7.19 Voigt Functions
See accompanying text
Figure 7.19.2: Voigt function V ( x , t ) , t = 0.1 , 2.5 , 5 , 10 . Magnify
lim t 0 V ( x , t ) = x 1 + x 2 .
V ( x , t ) = V ( x , t ) .
1 V ( x , t ) 1 .
7.19.11 V ( u a , 1 4 a 2 ) = a 0 e a t 1 4 t 2 sin ( u t ) d t .
3: 5.19 Mathematical Applications
The volume V and surface area S of the n -dimensional sphere of radius r are given by
V = π 1 2 n r n Γ ( 1 2 n + 1 ) ,
S = 2 π 1 2 n r n 1 Γ ( 1 2 n ) = n r V .
4: 18.39 Physical Applications
Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) :
18.39.1 ( 2 2 m 2 x 2 + V ( x ) ) ψ ( x , t ) = i t ψ ( x , t ) ,
18.39.2 d 2 η d x 2 + 2 m 2 ( E V ( x ) ) η = 0 .
18.39.3 V ( x ) = 1 2 m ω 2 x 2 ,
18.39.6 2 ψ + 2 m 2 ( E V ( x ) ) ψ = 0 ,
5: 12.3 Graphics
See accompanying text
Figure 12.3.2: V ( a , x ) , a = 0. … Magnify
See accompanying text
Figure 12.3.4: V ( a , x ) , a = 0.5 , 2 , 3.5 , 5 . Magnify
See accompanying text
Figure 12.3.8: V ( a , x ) , 2.5 a 2.5 , 2.5 x 2.5 . Magnify 3D Help
6: 36.13 Kelvin’s Ship-Wave Pattern
A ship moving with constant speed V on deep water generates a surface gravity wave. …
36.13.2 ρ = g r / V 2 .
36.13.6 ω ( k ) = g k + V k .
Here k = | k | , and V is the ship velocity (so that V = | V | ). …
7: 9.1 Special Notation
Other notations that have been used are as follows: Ai ( x ) and Bi ( x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 1 / 3 π Ai ( 3 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( x ) , e ~ 0 ( x ) = π Gi ( x ) (Tumarkin (1959)).
8: 13.28 Physical Applications
f 1 ( ξ ) = ξ 1 2 V κ , 1 2 p ( 1 ) ( 2 i k ξ ) ,
f 2 ( η ) = η 1 2 V κ , 1 2 p ( 2 ) ( 2 i k η ) ,
and V κ , μ ( j ) ( z ) , j = 1 , 2 , denotes any pair of solutions of Whittaker’s equation (13.14.1). …
9: 19.33 Triaxial Ellipsoids
The surface area of an ellipsoid with semiaxes a , b , c , and volume V = 4 π a b c / 3 is given by
19.33.1 S = 3 V R G ( a 2 , b 2 , c 2 ) ,
The potential is …and the electric capacity C = Q / V ( 0 ) is given by … Let a homogeneous magnetic ellipsoid with semiaxes a , b , c , volume V = 4 π a b c / 3 , and susceptibility χ be placed in a previously uniform magnetic field H parallel to the principal axis with semiaxis c . …
10: 28.32 Mathematical Applications
28.32.2 2 V x 2 + 2 V y 2 + k 2 V = 0
then becomes
28.32.3 2 V ξ 2 + 2 V η 2 + 1 2 c 2 k 2 ( cosh ( 2 ξ ) cos ( 2 η ) ) V = 0 .
The separated solutions V ( ξ , η ) = v ( ξ ) w ( η ) can be obtained from the modified Mathieu’s equation (28.20.1) for v and from Mathieu’s equation (28.2.1) for w , where a is the separation constant and q = 1 4 c 2 k 2 . …
28.32.8 2 V + k 2 V = 0