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derivatives of the error function

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1: 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 , .
2: 7.18 Repeated Integrals of the Complementary Error Function
§7.18(iii) Properties
7.18.3 d d z i n erfc ( z ) = - i n - 1 erfc ( z ) , n = 0 , 1 , 2 , ,
7.18.4 d n d z n ( e z 2 erfc z ) = ( - 1 ) n 2 n n ! e z 2 i n erfc ( z ) , n = 0 , 1 , 2 , .
7.18.5 d 2 W d z 2 + 2 z d W d z - 2 n W = 0 , W ( z ) = A i n erfc ( z ) + B i n erfc ( - z ) ,
3: 12.7 Relations to Other Functions
4: 10.21 Zeros
5: 7.21 Physical Applications
§7.21 Physical Applications
The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function i n erfc ( z ) . Fried and Conte (1961) mentions the role of w ( z ) in the theory of linearized waves or oscillations in a hot plasma; w ( z ) is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …
6: 7.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. …
7: 2.7 Differential Equations
2.7.24 F ( x ) = ( 1 f 1 / 4 d 2 d x 2 ( 1 f 1 / 4 ) - g f 1 / 2 ) d x ,
2.7.25 𝒱 a j , x ( F ) = a j x | ( 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) - g ( t ) f 1 / 2 ( t ) ) d t | .
8: 10.40 Asymptotic Expansions for Large Argument
ν -Derivative
§10.40(ii) Error Bounds for Real Argument and Order
For the error term in (10.40.1) see §10.40(iii).
§10.40(iii) Error Bounds for Complex Argument and Order
9: 13.3 Recurrence Relations and Derivatives
§13.3 Recurrence Relations and Derivatives
§13.3(i) Recurrence Relations
§13.3(ii) Differentiation Formulas
13.3.22 d d z U ( a , b , z ) = - a U ( a + 1 , b + 1 , z ) ,
13.3.29 ( z d d z z ) n = z n d n d z n z n , n = 1 , 2 , 3 , .
10: 10.63 Recurrence Relations and Derivatives
§10.63 Recurrence Relations and Derivatives
§10.63(i) ber ν x , bei ν x , ker ν x , kei ν x
Let f ν ( x ) , g ν ( x ) denote any one of the ordered pairs: …
§10.63(ii) Cross-Products