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1: 1.5 Calculus of Two or More Variables
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§1.5(i) Partial Derivatives
β–ΊThe 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. … β–ΊSufficient 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 Ξ± ⁑ ( x ) and Ξ² ⁑ ( x ) are continuously differentiable and lie in [ c , d ] . … β–ΊSuppose 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 ) . Suppose also that c d f ⁑ ( x , y ) ⁒ d y converges and c d ( f / x ) ⁒ d y converges uniformly on a x b , that is, given any positive number Ο΅ , however small, we can find a number c 0 [ c , d ) that is independent of x and is such that …
2: 19.18 Derivatives and Differential Equations
β–ΊLet j = / z j , and 𝐞 j be an n -tuple with 1 in the j th place and 0’s elsewhere. … β–ΊIf 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). … β–Ί
19.18.14 2 w x 2 = 2 w y 2 + 1 y ⁒ w y .
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19.18.15 2 W t 2 = 2 W x 2 + 2 W y 2 .
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19.18.16 2 u x 2 + 2 u y 2 + 1 y ⁒ u y = 0 ,
3: 36.10 Differential Equations
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§36.10(ii) Partial Derivatives with Respect to the x n
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36.10.10 3 ⁒ n Ψ 3 y 3 ⁒ n = i n ⁒ 2 ⁒ n Ψ 3 z 2 ⁒ n .
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§36.10(iv) Partial z -Derivatives
4: 16.14 Partial Differential Equations
§16.14 Partial Differential Equations
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§16.14(i) Appell Functions
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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 ,
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y ⁒ ( 1 y ) ⁒ 2 F 1 y 2 + x ⁒ ( 1 y ) ⁒ 2 F 1 x ⁒ y + ( γ ( α + β + 1 ) ⁒ y ) ⁒ F 1 y β ⁒ x ⁒ F 1 x α ⁒ β ⁒ F 1 = 0 ,
β–ΊIn 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. …
5: 10.38 Derivatives with Respect to Order
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10.38.2 K Ξ½ ⁑ ( z ) Ξ½ = 1 2 ⁒ Ο€ ⁒ csc ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ ( I Ξ½ ⁑ ( z ) Ξ½ I Ξ½ ⁑ ( z ) Ξ½ ) Ο€ ⁒ cot ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ K Ξ½ ⁑ ( z ) , Ξ½ β„€ .
β–ΊFor I Ξ½ ⁑ ( z ) / Ξ½ at Ξ½ = n combine (10.38.1), (10.38.2), and (10.38.4). β–Ί
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 ) .
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I ν ⁑ ( z ) ν | ν = 0 = K 0 ⁑ ( z ) ,
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6: 10.15 Derivatives with Respect to Order
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10.15.1 J ± Ξ½ ⁑ ( z ) Ξ½ = ± J ± Ξ½ ⁑ ( z ) ⁒ ln ⁑ ( 1 2 ⁒ z ) βˆ“ ( 1 2 ⁒ z ) ± Ξ½ ⁒ k = 0 ( 1 ) k ⁒ ψ ⁑ ( k + 1 ± Ξ½ ) Ξ“ ⁑ ( k + 1 ± Ξ½ ) ⁒ ( 1 4 ⁒ z 2 ) k k ! ,
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10.15.2 Y Ξ½ ⁑ ( z ) Ξ½ = cot ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ ( J Ξ½ ⁑ ( z ) Ξ½ Ο€ ⁒ Y Ξ½ ⁑ ( z ) ) csc ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ J Ξ½ ⁑ ( z ) Ξ½ Ο€ ⁒ J Ξ½ ⁑ ( z ) .
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10.15.3 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = n = Ο€ 2 ⁒ Y n ⁑ ( z ) + n ! 2 ⁒ ( 1 2 ⁒ z ) n ⁒ k = 0 n 1 ( 1 2 ⁒ z ) k ⁒ J k ⁑ ( z ) k ! ⁒ ( n k ) .
β–ΊFor J Ξ½ ⁑ ( z ) / Ξ½ at Ξ½ = n combine (10.2.4) and (10.15.3). … β–Ί
10.15.5 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ Y 0 ⁑ ( z ) , Y Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ J 0 ⁑ ( z ) .
7: 12.17 Physical Applications
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12.17.2 2 = 2 x 2 + 2 y 2 + 2 z 2
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12.17.4 1 ξ 2 + η 2 ⁒ ( 2 w ξ 2 + 2 w η 2 ) + 2 w ΢ 2 + k 2 ⁒ w = 0 .
8: 10.73 Physical Applications
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10.73.1 2 V = 1 r ⁒ r ⁑ ( r ⁒ V r ) + 1 r 2 ⁒ 2 V Ο• 2 + 2 V z 2 = 0 ,
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10.73.2 2 ψ = 1 c 2 ⁒ 2 ψ t 2 ,
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10.73.3 4 W + λ 2 ⁒ 2 W t 2 = 0 .
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10.73.4 ( 2 + k 2 ) ⁒ f = 1 ρ 2 ⁒ ρ ⁑ ( ρ 2 ⁒ f ρ ) + 1 ρ 2 ⁒ sin ⁑ ΞΈ ⁒ ΞΈ ⁑ ( sin ⁑ ΞΈ ⁒ f ΞΈ ) + 1 ρ 2 ⁒ sin 2 ⁑ ΞΈ ⁒ 2 f Ο• 2 + k 2 ⁒ f .
9: 23.21 Physical Applications
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§23.21(ii) Nonlinear Evolution Equations
β–ΊAirault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … β–Ί
23.21.2 ( η ΢ ) ⁒ ( ΢ ξ ) ⁒ ( ξ η ) ⁒ 2 = ( ΢ η ) ⁒ f ⁑ ( ξ ) ⁒ f ⁑ ( ξ ) ⁒ ξ + ( ξ ΢ ) ⁒ f ⁑ ( η ) ⁒ f ⁑ ( η ) ⁒ η + ( η ξ ) ⁒ f ⁑ ( ΢ ) ⁒ f ⁑ ( ΢ ) ⁒ ΢ ,
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23.21.5 ( ⁑ ( v ) ⁑ ( w ) ) ⁒ ( ⁑ ( w ) ⁑ ( u ) ) ⁒ ( ⁑ ( u ) ⁑ ( v ) ) ⁒ 2 = ( ⁑ ( w ) ⁑ ( v ) ) ⁒ 2 u 2 + ( ⁑ ( u ) ⁑ ( w ) ) ⁒ 2 v 2 + ( ⁑ ( v ) ⁑ ( u ) ) ⁒ 2 w 2 .
10: 36.4 Bifurcation Sets
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s ⁑ Φ ( U ) ⁑ ( s j ⁒ ( 𝐱 ) , t j ⁑ ( 𝐱 ) ; 𝐱 ) = 0 ,
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t ⁑ Φ ( U ) ⁑ ( s j ⁒ ( 𝐱 ) , t j ⁑ ( 𝐱 ) ; 𝐱 ) = 0 .
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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 .