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1: 36.10 Differential Equations
36.10.6 l n Ψ K x m l n = i n ( l - m ) m n Ψ K x l m n , 1 m K , 1 l K .
36.10.9 3 n Ψ 3 x 3 n = ( - 1 ) n n Ψ 3 z n ,
36.10.10 3 n Ψ 3 y 3 n = i n 2 n Ψ 3 z 2 n .
2: 10.38 Derivatives with Respect to Order
10.38.2 K ν ( z ) ν = 1 2 π csc ( ν π ) ( I - ν ( z ) ν - I ν ( z ) ν ) - π cot ( ν π ) K ν ( z ) , ν .
For I ν ( z ) / ν at ν = - n combine (10.38.1), (10.38.2), and (10.38.4).
10.38.4 K ν ( z ) ν | ν = n = n ! 2 ( 1 2 z ) n k = 0 n - 1 ( 1 2 z ) k K k ( z ) k ! ( n - k ) .
I ν ( z ) ν | ν = 0 = - K 0 ( z ) ,
K ν ( z ) ν | ν = 0 = 0 .
3: 19.18 Derivatives and Differential Equations
Let j = / z j , and e j be an n -tuple with 1 in the j th place and 0’s elsewhere. …
19.18.14 2 w x 2 = 2 w y 2 + 1 y w y .
19.18.15 2 W t 2 = 2 W x 2 + 2 W y 2 .
19.18.16 2 u x 2 + 2 u y 2 + 1 y u y = 0 ,
19.18.17 2 U x 2 + 2 U y 2 + 2 U z 2 = 0 .
4: 1.5 Calculus of Two or More Variables
1.5.8 u f ( x ( u , v ) , y ( u , v ) ) = f x x u + f y y u ,
1.5.9 v f ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) = f x x v + f y y v + f z z v .
1.5.19 f x = f y = 0  at  ( a , b ) ,
1.5.38 ( f , g ) ( x , y ) = | f / x f / y g / x g / y | ,
1.5.40 ( f , g , h ) ( x , y , z ) = | f / x f / y f / z g / x g / y g / z h / x h / y h / z | ,
5: 16.14 Partial Differential Equations
x ( 1 - x ) 2 F 1 x 2 + y ( 1 - x ) 2 F 1 x y + ( γ - ( α + β + 1 ) x ) F 1 x - β y F 1 y - α β F 1 = 0 ,
x ( 1 - x ) 2 F 2 x 2 - x y 2 F 2 x y + ( γ - ( α + β + 1 ) x ) F 2 x - β y F 2 y - α β F 2 = 0 ,
y ( 1 - y ) 2 F 2 y 2 - x y 2 F 2 x y + ( γ - ( α + β + 1 ) y ) F 2 y - β x F 2 x - α β F 2 = 0 ,
x ( 1 - x ) 2 F 4 x 2 - 2 x y 2 F 4 x y - y 2 2 F 4 y 2 + ( γ - ( α + β + 1 ) x ) F 4 x - ( α + β + 1 ) y F 4 y - α β F 4 = 0 ,
y ( 1 - y ) 2 F 4 y 2 - 2 x y 2 F 4 x y - x 2 2 F 4 x 2 + ( γ - ( α + β + 1 ) y ) F 4 y - ( α + β + 1 ) x F 4 x - α β F 4 = 0 .
6: 10.15 Derivatives with Respect to Order
10.15.1 J ± ν ( z ) ν = ± J ± ν ( z ) ln ( 1 2 z ) ( 1 2 z ) ± ν k = 0 ( - 1 ) k ψ ( k + 1 ± ν ) Γ ( k + 1 ± ν ) ( 1 4 z 2 ) k k ! ,
10.15.2 Y ν ( z ) ν = cot ( ν π ) ( J ν ( z ) ν - π Y ν ( z ) ) - csc ( ν π ) J - ν ( z ) ν - π J ν ( z ) .
10.15.3 J ν ( z ) ν | ν = n = π 2 Y n ( z ) + n ! 2 ( 1 2 z ) n k = 0 n - 1 ( 1 2 z ) k J k ( z ) k ! ( n - k ) .
For J ν ( z ) / ν at ν = - n combine (10.2.4) and (10.15.3). …
10.15.5 J ν ( z ) ν | ν = 0 = π 2 Y 0 ( z ) , Y ν ( z ) ν | ν = 0 = - π 2 J 0 ( z ) .
7: 10.73 Physical Applications
10.73.1 2 V = 1 r r ( r V r ) + 1 r 2 2 V ϕ 2 + 2 V z 2 = 0 ,
10.73.2 2 ψ = 1 c 2 2 ψ t 2 ,
10.73.3 4 W + λ 2 2 W t 2 = 0 .
10.73.4 ( 2 + k 2 ) f = 1 ρ 2 ρ ( ρ 2 f ρ ) + 1 ρ 2 sin θ θ ( sin θ f θ ) + 1 ρ 2 sin 2 θ 2 f ϕ 2 + k 2 f .
8: 12.17 Physical Applications
12.17.2 2 = 2 x 2 + 2 y 2 + 2 z 2
12.17.4 1 ξ 2 + η 2 ( 2 w ξ 2 + 2 w η 2 ) + 2 w ζ 2 + k 2 w = 0 .
9: 1.6 Vectors and Vector-Valued Functions
1.6.20 grad f = f = f x i + f y j + f z k .
1.6.21 div F = F = F 1 x + F 2 y + F 3 z .
1.6.46 T u = x u ( u 0 , v 0 ) i + y u ( u 0 , v 0 ) j + z u ( u 0 , v 0 ) k
1.6.47 T v = x v ( u 0 , v 0 ) i + y v ( u 0 , v 0 ) j + z v ( u 0 , v 0 ) k
where g / n = g n is the derivative of g normal to the surface outwards from V and n is the unit outer normal vector. …
10: 28.32 Mathematical Applications
28.32.2 2 V x 2 + 2 V y 2 + k 2 V = 0
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 .
28.32.4 2 K z 2 - 2 K ζ 2 = 2 q ( cos ( 2 z ) - cos ( 2 ζ ) ) K .
28.32.5 K ( z , ζ ) d u ( ζ ) d ζ - u ( ζ ) K ( z , ζ ) ζ