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31: 5.19 Mathematical Applications
32: 25.13 Periodic Zeta Function
25.13.2 F ( x , s ) = Γ ( 1 s ) ( 2 π ) 1 s ( e π i ( 1 s ) / 2 ζ ( 1 s , x ) + e π i ( s 1 ) / 2 ζ ( 1 s , 1 x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 s , x ) = Γ ( s ) ( 2 π ) s ( e π i s / 2 F ( x , s ) + e π i s / 2 F ( x , s ) ) , s > 0 if 0 < x < 1 ; s > 1 if x = 1 .
33: 10.49 Explicit Formulas
§10.49 Explicit Formulas
§10.49(i) Unmodified Functions
§10.49(ii) Modified Functions
§10.49(iii) Rayleigh’s Formulas
§10.49(iv) Sums or Differences of Squares
34: 14.9 Connection Formulas
§14.9 Connection Formulas
§14.9(i) Connections Between 𝖯 ν ± μ ( x ) , 𝖯 ν 1 ± μ ( x ) , 𝖰 ν ± μ ( x ) , 𝖰 ν 1 μ ( x )
§14.9(ii) Connections Between 𝖯 ν ± μ ( ± x ) , 𝖰 ν μ ( ± x ) , 𝖰 ν μ ( x )
§14.9(iii) Connections Between P ν ± μ ( x ) , P ν 1 ± μ ( x ) , 𝑸 ν ± μ ( x ) , 𝑸 ν 1 μ ( x )
§14.9(iv) Whipple’s Formula
35: 15.6 Integral Representations
15.6.1 𝐅 ( a , b ; c ; z ) = 1 Γ ( b ) Γ ( c b ) 0 1 t b 1 ( 1 t ) c b 1 ( 1 z t ) a d t , | ph ( 1 z ) | < π ; c > b > 0 .
15.6.2 𝐅 ( a , b ; c ; z ) = Γ ( 1 + b c ) 2 π i Γ ( b ) 0 ( 1 + ) t b 1 ( t 1 ) c b 1 ( 1 z t ) a d t , | ph ( 1 z ) | < π ; c b 1 , 2 , 3 , , b > 0 .
15.6.3 𝐅 ( a , b ; c ; z ) = e b π i Γ ( 1 b ) 2 π i Γ ( c b ) ( 0 + ) t b 1 ( t + 1 ) a c ( t z t + 1 ) a d t , | ph ( 1 z ) | < π ; b 1 , 2 , 3 , , ( c b ) > 0 .
15.6.4 𝐅 ( a , b ; c ; z ) = e b π i Γ ( 1 b ) 2 π i Γ ( c b ) 1 ( 0 + ) t b 1 ( 1 t ) c b 1 ( 1 z t ) a d t , | ph ( 1 z ) | < π ; b 1 , 2 , 3 , , ( c b ) > 0 .
15.6.8 𝐅 ( a , b ; c ; z ) = 1 Γ ( c d ) 0 1 𝐅 ( a , b ; d ; z t ) t d 1 ( 1 t ) c d 1 d t , | ph ( 1 z ) | < π ; c > d > 0 .
36: 3.4 Differentiation
Two-Point Formula
Three-Point Formula
Four-Point Formula
Five-Point Formula
Six-Point Formula
37: 36.5 Stokes Sets
§36.5(ii) Cuspoids
§36.5(iii) Umbilics
38: 10.4 Connection Formulas
§10.4 Connection Formulas
39: 6.18 Methods of Computation
Also, other ranges of ph z can be covered by use of the continuation formulas of §6.4. … For example, the Gauss–Laguerre formula3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula3.5(v)) see Tooper and Mark (1968). …
40: About MathML
MathML allows us to present the mathematics independent on your screen size and resolution, enabling adjustment to enlarge or shrink formula, as well as providing opportunities for making the material accessible for those with disabilities. …