About the Project

cross ratio

AdvancedHelp

(0.002 seconds)

1—10 of 34 matching pages

1: 1.9 Calculus of a Complex Variable
β–ΊThe cross ratio of z 1 , z 2 , z 3 , z 4 β„‚ { } is defined by …
2: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
β–Ί
10.28.1 𝒲 ⁑ { I Ξ½ ⁑ ( z ) , I Ξ½ ⁑ ( z ) } = I Ξ½ ⁑ ( z ) ⁒ I Ξ½ 1 ⁑ ( z ) I Ξ½ + 1 ⁑ ( z ) ⁒ I Ξ½ ⁑ ( z ) = 2 ⁒ sin ⁑ ( Ξ½ ⁒ Ο€ ) / ( Ο€ ⁒ z ) ,
3: 28.28 Integrals, Integral Representations, and Integral Equations
β–Ί
28.28.25 sinh ⁑ z Ο€ 2 ⁒ 0 2 ⁒ Ο€ cos ⁑ t ⁒ me Ξ½ ⁑ ( t , h 2 ) ⁒ me Ξ½ 2 ⁒ m 1 ⁑ ( t , h 2 ) sinh 2 ⁑ z + sin 2 ⁑ t ⁒ d t = ( 1 ) m + 1 ⁒ i ⁒ h ⁒ Ξ± Ξ½ , m ( 0 ) ⁒ D 0 ⁑ ( Ξ½ , Ξ½ + 2 ⁒ m + 1 , z ) ,
β–Ί
28.28.26 cosh ⁑ z Ο€ 2 ⁒ 0 2 ⁒ Ο€ sin ⁑ t ⁒ me Ξ½ ⁑ ( t , h 2 ) ⁒ me Ξ½ 2 ⁒ m 1 ⁑ ( t , h 2 ) sinh 2 ⁑ z + sin 2 ⁑ t ⁒ d t = ( 1 ) m + 1 ⁒ i ⁒ h ⁒ Ξ± Ξ½ , m ( 1 ) ⁒ D 0 ⁑ ( Ξ½ , Ξ½ + 2 ⁒ m + 1 , z ) ,
β–Ί
28.28.29 cosh ⁑ z Ο€ 2 ⁒ 0 2 ⁒ Ο€ sin ⁑ t ⁒ me Ξ½ ⁑ ( t , h 2 ) ⁒ me Ξ½ 2 ⁒ m 1 ⁑ ( t , h 2 ) sinh 2 ⁑ z + sin 2 ⁑ t ⁒ d t = ( 1 ) m + 1 ⁒ i ⁒ h ⁒ Ξ± Ξ½ , m ( 0 ) ⁒ D 1 ⁑ ( Ξ½ , Ξ½ + 2 ⁒ m + 1 , z ) ,
β–Ί
28.28.30 sinh ⁑ z Ο€ 2 ⁒ 0 2 ⁒ Ο€ cos ⁑ t ⁒ me Ξ½ ⁑ ( t , h 2 ) ⁒ me Ξ½ 2 ⁒ m 1 ⁑ ( t , h 2 ) sinh 2 ⁑ z + sin 2 ⁑ t ⁒ d t = ( 1 ) m ⁒ i ⁒ h ⁒ Ξ± Ξ½ , m ( 1 ) ⁒ D 1 ⁑ ( Ξ½ , Ξ½ + 2 ⁒ m + 1 , z ) ,
β–Ί
28.28.49 Ξ± ^ n , m ( c ) = 1 2 ⁒ Ο€ ⁒ 0 2 ⁒ Ο€ cos ⁑ t ⁒ ce n ⁑ ( t , h 2 ) ⁒ ce m ⁑ ( t , h 2 ) ⁒ d t = ( 1 ) p + 1 ⁒ 2 i ⁒ Ο€ ⁒ ce n ⁑ ( 0 , h 2 ) ⁒ ce m ⁑ ( 0 , h 2 ) h ⁒ Dc 0 ⁑ ( n , m , 0 ) .
4: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
β–Ί
10.5.1 𝒲 ⁑ { J Ξ½ ⁑ ( z ) , J Ξ½ ⁑ ( z ) } = J Ξ½ + 1 ⁑ ( z ) ⁒ J Ξ½ ⁑ ( z ) + J Ξ½ ⁑ ( z ) ⁒ J Ξ½ 1 ⁑ ( z ) = 2 ⁒ sin ⁑ ( Ξ½ ⁒ Ο€ ) / ( Ο€ ⁒ z ) ,
β–Ί
10.5.2 𝒲 ⁑ { J Ξ½ ⁑ ( z ) , Y Ξ½ ⁑ ( z ) } = J Ξ½ + 1 ⁑ ( z ) ⁒ Y Ξ½ ⁑ ( z ) J Ξ½ ⁑ ( z ) ⁒ Y Ξ½ + 1 ⁑ ( z ) = 2 / ( Ο€ ⁒ z ) ,
β–Ί
10.5.3 𝒲 ⁑ { J Ξ½ ⁑ ( z ) , H Ξ½ ( 1 ) ⁑ ( z ) } = J Ξ½ + 1 ⁑ ( z ) ⁒ H Ξ½ ( 1 ) ⁑ ( z ) J Ξ½ ⁑ ( z ) ⁒ H Ξ½ + 1 ( 1 ) ⁑ ( z ) = 2 ⁒ i / ( Ο€ ⁒ z ) ,
β–Ί
10.5.4 𝒲 ⁑ { J Ξ½ ⁑ ( z ) , H Ξ½ ( 2 ) ⁑ ( z ) } = J Ξ½ + 1 ⁑ ( z ) ⁒ H Ξ½ ( 2 ) ⁑ ( z ) J Ξ½ ⁑ ( z ) ⁒ H Ξ½ + 1 ( 2 ) ⁑ ( z ) = 2 ⁒ i / ( Ο€ ⁒ z ) ,
5: 10.6 Recurrence Relations and Derivatives
β–Ί
10.6.10 p Ξ½ ⁒ s Ξ½ q Ξ½ ⁒ r Ξ½ = 4 / ( Ο€ 2 ⁒ a ⁒ b ) .
6: 10.67 Asymptotic Expansions for Large Argument
β–Ί
10.67.1 ker Ξ½ ⁑ x e x / 2 ⁒ ( Ο€ 2 ⁒ x ) 1 2 ⁒ k = 0 a k ⁑ ( Ξ½ ) x k ⁒ cos ⁑ ( x 2 + ( Ξ½ 2 + k 4 + 1 8 ) ⁒ Ο€ ) ,
β–Ί
10.67.2 kei Ξ½ ⁑ x e x / 2 ⁒ ( Ο€ 2 ⁒ x ) 1 2 ⁒ k = 0 a k ⁑ ( Ξ½ ) x k ⁒ sin ⁑ ( x 2 + ( Ξ½ 2 + k 4 + 1 8 ) ⁒ Ο€ ) .
β–Ί
10.67.5 ker Ξ½ ⁑ x e x / 2 ⁒ ( Ο€ 2 ⁒ x ) 1 2 ⁒ k = 0 b k ⁑ ( Ξ½ ) x k ⁒ cos ⁑ ( x 2 + ( Ξ½ 2 + k 4 1 8 ) ⁒ Ο€ ) ,
β–Ί
10.67.6 kei Ξ½ ⁑ x e x / 2 ⁒ ( Ο€ 2 ⁒ x ) 1 2 ⁒ k = 0 b k ⁑ ( Ξ½ ) x k ⁒ sin ⁑ ( x 2 + ( Ξ½ 2 + k 4 1 8 ) ⁒ Ο€ ) .
β–Ί
§10.67(ii) Cross-Products and Sums of Squares in the Case Ξ½ = 0
7: 14.2 Differential Equations
β–Ί
14.2.11 P Ξ½ + 1 ΞΌ ⁑ ( x ) ⁒ Q Ξ½ ΞΌ ⁑ ( x ) P Ξ½ ΞΌ ⁑ ( x ) ⁒ Q Ξ½ + 1 ΞΌ ⁑ ( x ) = e ΞΌ ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) Ξ“ ⁑ ( Ξ½ ΞΌ + 2 ) .
8: 4.2 Definitions
β–Ί β–Ίwhere k is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. … β–Ί
4.2.20 exp ⁑ ( z + 2 ⁒ Ο€ ⁒ i ) = exp ⁑ z .
β–Ί
4.2.23 ph ⁑ ( exp ⁑ z ) = ⁑ z + 2 ⁒ k ⁒ Ο€ , k β„€ .
β–Ί
4.2.33 e z = ( exp ⁑ z ) ⁒ exp ⁑ ( 2 ⁒ k ⁒ z ⁒ Ο€ ⁒ i ) , k β„€ .
9: 13.4 Integral Representations
β–Ί
13.4.4 U ⁑ ( a , b , z ) = 1 Ξ“ ⁑ ( a ) ⁒ 0 e z ⁒ t ⁒ t a 1 ⁒ ( 1 + t ) b a 1 ⁒ d t , ⁑ a > 0 , | ph ⁑ z | < 1 2 ⁒ Ο€ ,
β–Ί
13.4.5 U ⁑ ( a , b , z ) = z 1 a Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( 1 + a b ) ⁒ 0 U ⁑ ( b a , b , t ) ⁒ e t ⁒ t a 1 t + z ⁒ d t , | ph ⁑ z | < Ο€ , ⁑ a > max ⁑ ( ⁑ b 1 , 0 ) ,
β–Ί
13.4.9 𝐌 ⁑ ( a , b , z ) = Ξ“ ⁑ ( 1 + a b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ 0 ( 1 + ) e z ⁒ t ⁒ t a 1 ⁒ ( t 1 ) b a 1 ⁒ d t , b a 1 , 2 , 3 , , ⁑ a > 0 .
β–ΊAt the point where the contour crosses the interval ( 1 , ) , t b and the 𝐅 1 2 function assume their principal values; compare §§15.1 and 15.2(i). … β–Ί
13.4.13 𝐌 ⁑ ( a , b , z ) = z 1 b 2 ⁒ Ο€ ⁒ i ⁒ ( 0 + , 1 + ) e z ⁒ t ⁒ t b ⁒ ( 1 1 t ) a ⁒ d t , | ph ⁑ z | < 1 2 ⁒ Ο€ .
10: 8.19 Generalized Exponential Integral
β–ΊWhen the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of E p ⁑ ( z ) , and unless indicated otherwise in the DLMF principal values are assumed. … β–Ί
8.19.3 E p ⁑ ( z ) = 1 e z ⁒ t t p ⁒ d t , | ph ⁑ z | < 1 2 ⁒ Ο€ ,
β–Ί
8.19.4 E p ⁑ ( z ) = z p 1 ⁒ e z Ξ“ ⁑ ( p ) ⁒ 0 t p 1 ⁒ e z ⁒ t 1 + t ⁒ d t , | ph ⁑ z | < 1 2 ⁒ Ο€ , ⁑ p > 0 .
β–Ί
8.19.17 E p ⁑ ( z ) = e z ⁒ ( 1 z + p 1 + 1 z + p + 1 1 + 2 z + β‹― ) , | ph ⁑ z | < Ο€ .
β–Ί
8.19.23 z E p 1 ⁑ ( t ) ⁒ d t = E p ⁑ ( z ) , | ph ⁑ z | < Ο€ ,