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1: 1.4 Calculus of One Variable
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Functions of Bounded Variation
►►Lastly, whether or not the real numbers a and b satisfy a < b , and whether or not they are finite, we define 𝒱 a , b ⁡ ( f ) by (1.4.34) whenever this integral exists. This definition also applies when f ⁡ ( x ) is a complex function of the real variable x . …
2: 10.40 Asymptotic Expansions for Large Argument
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10.40.6 I ν ⁡ ( z ) ⁢ K ν ⁡ ( z ) 1 2 ⁢ z ⁢ ( 1 1 2 ⁢ μ 1 ( 2 ⁢ z ) 2 + 1 3 2 4 ⁢ ( μ 1 ) ⁢ ( μ 9 ) ( 2 ⁢ z ) 4 ⋯ ) ,
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10.40.7 I ν ⁡ ( z ) ⁢ K ν ⁡ ( z ) 1 2 ⁢ z ⁢ ( 1 + 1 2 ⁢ μ 3 ( 2 ⁢ z ) 2 1 2 4 ⁢ ( μ 1 ) ⁢ ( μ 45 ) ( 2 ⁢ z ) 4 + ⋯ ) ,
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§10.40(iii) Error Bounds for Complex Argument and Order
►where 𝒱 denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that | ⁡ t | changes monotonically. … ►
10.40.12 𝒱 z , ⁡ ( t ℓ ) { | z | ℓ , | ph ⁡ z | 1 2 ⁢ π , χ ⁡ ( ℓ ) ⁢ | z | ℓ , 1 2 ⁢ π | ph ⁡ z | π , 2 ⁢ χ ⁡ ( ℓ ) ⁢ | ⁡ z | ℓ , π | ph ⁡ z | < 3 2 ⁢ π ,
3: 8.23 Statistical Applications
§8.23 Statistical Applications
4: 10.77 Software
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§10.77(iv) Bessel Functions–Integer or Half-Integer Order and Complex Arguments, including Kelvin Functions
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§10.77(v) Bessel FunctionsReal Order and Complex Argument (including Hankel Functions)
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§10.77(vii) Bessel FunctionsComplex Order and Real Argument
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§10.77(viii) Bessel FunctionsComplex Order and Argument
5: 14.31 Other Applications
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§14.31(iii) Miscellaneous
►Legendre functions P ν ⁡ ( x ) of complex degree ν appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …
6: 30.6 Functions of Complex Argument
§30.6 Functions of Complex Argument
►The solutions … ►
30.6.3 𝒲 ⁡ { 𝑃𝑠 n m ⁡ ( z , γ 2 ) , 𝑄𝑠 n m ⁡ ( z , γ 2 ) } = ( 1 ) m ⁢ ( n + m ) ! ( 1 z 2 ) ⁢ ( n m ) ! ⁢ A n m ⁡ ( γ 2 ) ⁢ A n m ⁡ ( γ 2 ) ,
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7: 12.8 Recurrence Relations and Derivatives
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12.8.1 z ⁢ U ⁡ ( a , z ) U ⁡ ( a 1 , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 ,
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12.8.2 U ⁡ ( a , z ) + 1 2 ⁢ z ⁢ U ⁡ ( a , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 ,
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12.8.3 U ⁡ ( a , z ) 1 2 ⁢ z ⁢ U ⁡ ( a , z ) + U ⁡ ( a 1 , z ) = 0 ,
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12.8.4 2 ⁢ U ⁡ ( a , z ) + U ⁡ ( a 1 , z ) + ( a + 1 2 ) ⁢ U ⁡ ( a + 1 , z ) = 0 .
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12.8.5 z ⁢ V ⁡ ( a , z ) V ⁡ ( a + 1 , z ) + ( a 1 2 ) ⁢ V ⁡ ( a 1 , z ) = 0 ,
8: 5.23 Approximations
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§5.23(iii) Approximations in the Complex Plane
9: 22.17 Moduli Outside the Interval [0,1]
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22.17.1 p ⁣ q ⁡ ( z , k ) = p ⁣ q ⁡ ( z , k ) ,
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22.17.2 sn ⁡ ( z , 1 / k ) = k ⁢ sn ⁡ ( z / k , k ) ,
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22.17.3 cn ⁡ ( z , 1 / k ) = dn ⁡ ( z / k , k ) ,
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22.17.4 dn ⁡ ( z , 1 / k ) = cn ⁡ ( z / k , k ) .
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§22.17(ii) Complex Moduli
10: 4.3 Graphics
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§4.3(ii) Complex Arguments: Conformal Maps
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See accompanying text
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Figure 4.3.3: ln ⁡ ( x + i ⁢ y ) (principal value). … Magnify 3D Help
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See accompanying text
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Figure 4.3.4: e x + i ⁢ y . Magnify 3D Help