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1: 4.20 Derivatives and Differential Equations
§4.20 Derivatives and Differential Equations
4.20.1 d d z sin z = cos z ,
4.20.2 d d z cos z = - sin z ,
4.20.3 d d z tan z = sec 2 z ,
4.20.9 d 2 w d z 2 + a 2 w = 0 ,
2: 4.34 Derivatives and Differential Equations
§4.34 Derivatives and Differential Equations
4.34.7 d 2 w d z 2 - a 2 w = 0 ,
4.34.8 ( d w d z ) 2 - a 2 w 2 = 1 ,
4.34.9 ( d w d z ) 2 - a 2 w 2 = - 1 ,
4.34.10 d w d z + a 2 w 2 = 1 ,
3: 4.7 Derivatives and Differential Equations
§4.7 Derivatives and Differential Equations
§4.7(i) Logarithms
4.7.1 d d z ln z = 1 z ,
4.7.5 d w d z = f ( z ) f ( z )
§4.7(ii) Exponentials and Powers
4: 36.10 Differential Equations
§36.10(ii) Partial Derivatives with Respect to the x n
§36.10(iv) Partial z -Derivatives
5: 19.18 Derivatives and Differential Equations
§19.18 Derivatives and Differential Equations
§19.18(i) Derivatives
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 .
6: 10.38 Derivatives with Respect to Order
§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). …
I ν ( z ) ν | ν = 0 = - K 0 ( z ) ,
K ν ( z ) ν | ν = 0 = 0 .
7: 7.10 Derivatives
§7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( - 1 ) n 2 π H n ( z ) e - z 2 , n = 0 , 1 , 2 , .
d f ( z ) d z = - π z g ( z ) ,
d g ( z ) d z = π z f ( z ) - 1 .
8: 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. …
Chain Rule
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 ) . …
§1.5(vi) Jacobians and Change of Variables
9: 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 .
d 2 U d ξ 2 + ( σ ξ 2 + λ ) U = 0 ,
d 2 V d η 2 + ( σ η 2 - λ ) V = 0 ,
d 2 W d ζ 2 + ( k 2 - σ ) W = 0 ,
10: 22.13 Derivatives and Differential Equations
§22.13 Derivatives and Differential Equations
§22.13(i) Derivatives
Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.
d d z ( sn z ) = cn z dn z d d z ( dc z )  = k 2 sc z nc z
Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. …
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 - sn 2 ( z , k ) ) ( 1 - k 2 sn 2 ( z , k ) ) ,