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partial differentiation

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1: 1.5 Calculus of Two or More Variables
2 f y x = y ( f x ) .
1.5.24 d d x c d f ( x , y ) d y = c d f x d y , a < x < b .
2: 3.4 Differentiation
§3.4(iii) Partial Derivatives
3: 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
4: 25.11 Hurwitz Zeta Function
25.11.17 a ζ ( s , a ) = - s ζ ( s + 1 , a ) , s 0 , 1 ; a > 0 .
5: 2.1 Definitions and Elementary Properties
means that for each n , the difference between f ( x ) and the n th partial sum on the right-hand side is O ( ( x - c ) n ) as x c in X . …
6: 1.9 Calculus of a Complex Variable
§1.9(ii) Continuity, Point Sets, and Differentiation
Differentiation
Differentiability automatically implies continuity.
Cauchy–Riemann Equations
Lastly, a power series can be differentiated any number of times within its circle of convergence: …
7: 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 .
8: 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 ) .
9: 11.10 Anger–Weber Functions
10: 5.19 Mathematical Applications
By decomposition into partial fractions (§1.2(iii)) …