About the Project
NIST

Weierstrass sigma function

AdvancedHelp

(0.004 seconds)

1—10 of 14 matching pages

1: 23.1 Special Notation
β–ΊThe main functions treated in this chapter are the Weierstrass -function ⁑ ( z ) = ⁑ ( z | 𝕃 ) = ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) ; the Weierstrass zeta function ΞΆ ⁑ ( z ) = ΞΆ ⁑ ( z | 𝕃 ) = ΞΆ ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) ; the Weierstrass sigma function Οƒ ⁑ ( z ) = Οƒ ⁑ ( z | 𝕃 ) = Οƒ ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) ; the elliptic modular function Ξ» ⁑ ( Ο„ ) ; Klein’s complete invariant J ⁑ ( Ο„ ) ; Dedekind’s eta function Ξ· ⁑ ( Ο„ ) . …
2: 23.2 Definitions and Periodic Properties
β–Ί
§23.2(ii) Weierstrass Elliptic Functions
β–Ί
23.2.6 Οƒ ⁑ ( z ) = z ⁒ w 𝕃 βˆ– { 0 } ( ( 1 - z w ) ⁒ exp ⁑ ( z w + z 2 2 ⁒ w 2 ) ) .
β–ΊThe function Οƒ ⁑ ( z ) is entire and odd, with simple zeros at the lattice points. … β–ΊFor j = 1 , 2 , 3 , the function Οƒ ⁑ ( z ) satisfies … β–ΊFor further quasi-periodic properties of the Οƒ -function see Lawden (1989, §6.2).
3: 23.10 Addition Theorems and Other Identities
β–Ί
23.10.3 Οƒ ⁑ ( u + v ) ⁒ Οƒ ⁑ ( u - v ) Οƒ 2 ⁑ ( u ) ⁒ Οƒ 2 ⁑ ( v ) = ⁑ ( v ) - ⁑ ( u ) ,
β–Ί
23.10.4 Οƒ ⁑ ( u + v ) ⁒ Οƒ ⁑ ( u - v ) ⁒ Οƒ ⁑ ( x + y ) ⁒ Οƒ ⁑ ( x - y ) + Οƒ ⁑ ( v + x ) ⁒ Οƒ ⁑ ( v - x ) ⁒ Οƒ ⁑ ( u + y ) ⁒ Οƒ ⁑ ( u - y ) + Οƒ ⁑ ( x + u ) ⁒ Οƒ ⁑ ( x - u ) ⁒ Οƒ ⁑ ( v + y ) ⁒ Οƒ ⁑ ( v - y ) = 0 .
β–ΊFor further addition-type identities for the Οƒ -function see Lawden (1989, §6.4). … β–Ί β–Ί
23.10.19 Οƒ ⁑ ( c ⁒ z | c ⁒ 𝕃 ) = c ⁒ Οƒ ⁑ ( z | 𝕃 ) .
4: 23.6 Relations to Other Functions
β–Ί
23.6.9 Οƒ ⁑ ( z ) = 2 ⁒ Ο‰ 1 ⁒ exp ⁑ ( Ξ· 1 ⁒ z 2 2 ⁒ Ο‰ 1 ) ⁒ ΞΈ 1 ⁑ ( Ο€ ⁒ z / ( 2 ⁒ Ο‰ 1 ) , q ) Ο€ ⁒ ΞΈ 1 ⁑ ( 0 , q ) ,
β–Ί β–Ί
23.6.12 Οƒ ⁑ ( Ο‰ 3 ) = - 2 ⁒ Ο‰ 1 ⁒ exp ⁑ ( 1 2 ⁒ Ξ· 1 ⁒ Ο‰ 1 ) ⁒ ΞΈ 4 ⁑ ( 0 , q ) Ο€ ⁒ q 1 / 4 ⁒ ΞΈ 1 ⁑ ( 0 , q ) .
β–ΊFor further results for the Οƒ -function see Lawden (1989, §6.2). … β–ΊFor representations of the Jacobi functions sn , cn , and dn as quotients of Οƒ -functions see Lawden (1989, §§6.2, 6.3). …
5: 23.3 Differential Equations
β–ΊAs functions of g 2 ⁑ and g 3 ⁑ , ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) and ΞΆ ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) are meromorphic and Οƒ ⁑ ( z ; g 2 ⁑ , g 3 ⁑ ) is entire. …
6: 23.4 Graphics
β–ΊLine graphs of the Weierstrass functions ⁑ ( x ) , ΞΆ ⁑ ( x ) , and Οƒ ⁑ ( x ) , illustrating the lemniscatic and equianharmonic cases. … β–ΊSurfaces for the Weierstrass functions ⁑ ( z ) , ΞΆ ⁑ ( z ) , and Οƒ ⁑ ( z ) . …
7: 23.9 Laurent and Other Power Series
β–Ί
23.9.7 Οƒ ⁑ ( z ) = m , n = 0 a m , n ⁒ ( 10 ⁒ c 2 ) m ⁒ ( 56 ⁒ c 3 ) n ⁒ z 4 ⁒ m + 6 ⁒ n + 1 ( 4 ⁒ m + 6 ⁒ n + 1 ) ! ,
8: 23.12 Asymptotic Approximations
β–Ί
23.12.3 Οƒ ⁑ ( z ) = 2 ⁒ Ο‰ 1 Ο€ ⁒ exp ⁑ ( Ο€ 2 ⁒ z 2 24 ⁒ Ο‰ 1 2 ) ⁒ sin ⁑ ( Ο€ ⁒ z 2 ⁒ Ο‰ 1 ) ⁒ ( 1 - ( Ο€ 2 ⁒ z 2 Ο‰ 1 2 - 4 ⁒ sin 2 ⁑ ( Ο€ ⁒ z 2 ⁒ Ο‰ 1 ) ) ⁒ q 2 + O ⁑ ( q 4 ) ) ,
9: 23.8 Trigonometric Series and Products
β–Ί
23.8.6 Οƒ ⁑ ( z ) = 2 ⁒ Ο‰ 1 Ο€ ⁒ exp ⁑ ( Ξ· 1 ⁒ z 2 2 ⁒ Ο‰ 1 ) ⁒ sin ⁑ ( Ο€ ⁒ z 2 ⁒ Ο‰ 1 ) ⁒ n = 1 1 - 2 ⁒ q 2 ⁒ n ⁒ cos ⁑ ( Ο€ ⁒ z / Ο‰ 1 ) + q 4 ⁒ n ( 1 - q 2 ⁒ n ) 2 ,
β–Ί
23.8.7 Οƒ ⁑ ( z ) = 2 ⁒ Ο‰ 1 Ο€ ⁒ exp ⁑ ( Ξ· 1 ⁒ z 2 2 ⁒ Ο‰ 1 ) ⁒ sin ⁑ ( Ο€ ⁒ z 2 ⁒ Ο‰ 1 ) ⁒ n = 1 sin ⁑ ( Ο€ ⁒ ( 2 ⁒ n ⁒ Ο‰ 3 + z ) / ( 2 ⁒ Ο‰ 1 ) ) ⁒ sin ⁑ ( Ο€ ⁒ ( 2 ⁒ n ⁒ Ο‰ 3 - z ) / ( 2 ⁒ Ο‰ 1 ) ) sin 2 ⁑ ( Ο€ ⁒ n ⁒ Ο‰ 3 / Ο‰ 1 ) .
10: 19.25 Relations to Other Functions
β–Ί
19.25.40 z = Οƒ ⁑ ( z ) ⁒ R F ⁑ ( Οƒ 1 2 ⁑ ( z ) , Οƒ 2 2 ⁑ ( z ) , Οƒ 3 2 ⁑ ( z ) ) ,
β–Ί
19.25.41 Οƒ j ⁑ ( z ) = exp ⁑ ( - Ξ· j ⁒ z ) ⁒ Οƒ ⁑ ( z + Ο‰ j ) / Οƒ ⁑ ( Ο‰ j ) , j = 1 , 2 , 3 .