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1: 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. … If n = 2 , then elimination of 2 v between (19.18.11) and (19.18.12), followed by the substitution ( b 1 , b 2 , z 1 , z 2 ) = ( b , c - b , 1 - z , 1 ) , produces the Gauss hypergeometric equation (15.10.1). …
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 ,
2: 36.10 Differential Equations
§36.10(ii) Partial Derivatives with Respect to the x n
36.10.10 3 n Ψ 3 y 3 n = i n 2 n Ψ 3 z 2 n .
§36.10(iv) Partial z -Derivatives
3: 16.14 Partial Differential Equations
§16.14 Partial Differential Equations
§16.14(i) Appell Functions
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 ,
In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F 1 2 functions, and which satisfy pairs of linear partial differential equations of the second order. …
4: 10.38 Derivatives with Respect to Order
10.38.1 I ± ν ( z ) ν = ± I ± ν ( z ) ln ( 1 2 z ) ( 1 2 z ) ± ν k = 0 ψ ( k + 1 ± ν ) Γ ( k + 1 ± ν ) ( 1 4 z 2 ) k k ! ,
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 ) .
K ν ( z ) ν | ν = 0 = 0 .
5: 1.5 Calculus of Two or More Variables
§1.5(i) Partial Derivatives
The function f ( x , y ) is continuously differentiable if f , f / x , and f / y are continuous, and twice-continuously differentiable if also 2 f / x 2 , 2 f / y 2 , 2 f / x y , and 2 f / y x are continuous. … If F ( x , y ) is continuously differentiable, F ( a , b ) = 0 , and F / y 0 at ( a , b ) , then in a neighborhood of ( a , b ) , that is, an open disk centered at a , b , the equation F ( x , y ) = 0 defines a continuously differentiable function y = g ( x ) such that F ( x , g ( x ) ) = 0 , b = g ( a ) , and g ( x ) = - F x / F y . … Sufficient conditions for validity are: (a) f and f / x are continuous on a rectangle a x b , c y d ; (b) when x [ a , b ] both α ( x ) and β ( x ) are continuously differentiable and lie in [ c , d ] . … Suppose that a , b , c are finite, d is finite or + , and f ( x , y ) , f / x are continuous on the partly-closed rectangle or infinite strip [ a , b ] × [ c , d ) . …
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: 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
10: 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
Suppose S is an oriented surface with boundary S which is oriented so that its direction is clockwise relative to the normals of S . … 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. …