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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
2: 3.4 Differentiation
§3.4 Differentiation
and follows from the differentiated form of (3.3.4). …
§3.4(ii) Analytic Functions
3: 2.1 Definitions and Elementary Properties
§2.1(ii) Integration and Differentiation
Differentiation requires extra conditions. …This result also holds with both O ’s replaced by o ’s. … 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 . … Differentiation, however, requires the kind of extra conditions needed for the O symbol (§2.1(ii)). …
4: 7.10 Derivatives
5: 9.10 Integrals
9.10.1 z Ai ( t ) d t = π ( Ai ( z ) Gi ( z ) Ai ( z ) Gi ( z ) ) ,
9.10.8 z w ( z ) d z = w ( z ) ,
9.10.9 z 2 w ( z ) d z = z w ( z ) w ( z ) ,
9.10.10 z n + 3 w ( z ) d z = z n + 2 w ( z ) ( n + 2 ) z n + 1 w ( z ) + ( n + 1 ) ( n + 2 ) z n w ( z ) d z , n = 0 , 1 , 2 , .
9.10.20 0 x 0 v Ai ( t ) d t d v = x 0 x Ai ( t ) d t Ai ( x ) + Ai ( 0 ) ,
6: 1.5 Calculus of Two or More Variables
2 f y x = y ( f x ) .
§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
1.5.24 d d x c d f ( x , y ) d y = c d f x d y , a < x < b .
7: 1.8 Fourier Series
§1.8(iii) Integration and Differentiation
If a function f ( x ) C 2 [ 0 , 2 π ] is periodic, with period 2 π , then the series obtained by differentiating the Fourier series for f ( x ) term by term converges at every point to f ( x ) . …
8: 4.10 Integrals
9: 5.19 Mathematical Applications
10: 10.57 Uniform Asymptotic Expansions for Large Order