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1: 3.4 Differentiation
§3.4 Differentiation
§3.4(iii) Partial Derivatives
The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). …
2: 36.10 Differential Equations
§36.10(ii) Partial Derivatives with Respect to the x n
§36.10(iv) Partial z -Derivatives
3: 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. …
§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
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 ) . …
4: 10.38 Derivatives with Respect to Order
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 ) ,
5: 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 ) .
6: 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 ,
See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …
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 .
7: 36.4 Bifurcation Sets
s Φ ( U ) ( s j ( 𝐱 ) , t j ( 𝐱 ) ; 𝐱 ) = 0 ,
t Φ ( U ) ( s j ( 𝐱 ) , t j ( 𝐱 ) ; 𝐱 ) = 0 .
36.4.4 2 s 2 Φ ( U ) ( s , t ; 𝐱 ) 2 t 2 Φ ( U ) ( s , t ; 𝐱 ) ( 2 s t Φ ( U ) ( s , t ; 𝐱 ) ) 2 = 0 .
8: 19.4 Derivatives and Differential Equations
Let D k = / k . …
9: 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 ,
10: 24.20 Tables
Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. In Wagstaff (2002) these results are extended to n = 60 ( 2 ) 152 and n = 40 ( 2 ) 88 , respectively, with further complete and partial factorizations listed up to n = 300 and n = 200 , respectively. …