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1: 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 ,
4.20.6 d d z cot z = csc 2 z ,
4.20.9 d 2 w d z 2 + a 2 w = 0 ,
2: 4.34 Derivatives and Differential Equations
4.34.1 d d z sinh z = cosh z ,
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: 22.13 Derivatives and Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
22.13.2 ( d d z cn ( z , k ) ) 2 = ( 1 cn 2 ( z , k ) ) ( k 2 + k 2 cn 2 ( z , k ) ) ,
22.13.3 ( d d z dn ( z , k ) ) 2 = ( 1 dn 2 ( z , k ) ) ( dn 2 ( z , k ) k 2 ) .
22.13.7 ( d d z dc ( z , k ) ) 2 = ( dc 2 ( z , k ) 1 ) ( dc 2 ( z , k ) k 2 ) ,
22.13.10 ( d d z ns ( z , k ) ) 2 = ( ns 2 ( z , k ) k 2 ) ( ns 2 ( z , k ) 1 ) ,
4: 4.7 Derivatives and Differential Equations
4.7.1 d d z ln z = 1 z ,
4.7.5 d w d z = f ( z ) f ( z )
4.7.7 d d z e z = e z ,
4.7.10 d d z z a = a z a 1 ,
4.7.12 d w d z = f ( z ) w
5: 27.6 Divisor Sums
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
27.6.2 d | n μ ( d ) f ( d ) = p | n ( 1 f ( p ) ) , n > 1 .
27.6.6 d | n ϕ k ( d ) ( n d ) k = 1 k + 2 k + + n k ,
27.6.7 d | n μ ( d ) ( n d ) k = J k ( n ) ,
27.6.8 d | n J k ( d ) = n k .
6: 31 Heun Functions
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7: 27.5 Inversion Formulas
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
27.5.2 d | n μ ( d ) = 1 n ,
27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
27.5.4 n = d | n ϕ ( d ) ϕ ( n ) = d | n d μ ( n d ) ,
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
8: Brian D. Sleeman
Profile
Brian D. Sleeman
Brian D. Sleeman (b. …, d. … D. …D. …
9: 31.12 Confluent Forms of Heun’s Equation
31.12.1 d 2 w d z 2 + ( γ z + δ z 1 + ϵ ) d w d z + α z q z ( z 1 ) w = 0 .
31.12.2 d 2 w d z 2 + ( δ z 2 + γ z + 1 ) d w d z + α z q z 2 w = 0 .
31.12.3 d 2 w d z 2 ( γ z + δ + z ) d w d z + α z q z w = 0 .
31.12.4 d 2 w d z 2 + ( γ + z ) z d w d z + ( α z q ) w = 0 .
For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …
10: 30.12 Generalized and Coulomb Spheroidal Functions
30.12.1 d d z ( ( 1 z 2 ) d w d z ) + ( λ + α z + γ 2 ( 1 z 2 ) μ 2 1 z 2 ) w = 0 ,
30.12.2 d d z ( ( 1 z 2 ) d w d z ) + ( λ + γ 2 ( 1 z 2 ) α ( α + 1 ) z 2 μ 2 1 z 2 ) w = 0 ,