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1: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
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q -Binomial Theorem
2: 1.2 Elementary Algebra
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Binomial Theorem
3: 17.2 Calculus
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§17.2(iii) Binomial Theorem
►In the limit as q 1 , (17.2.35) reduces to the standard binomial theorem
4: 4.38 Inverse Hyperbolic Functions: Further Properties
5: 18.18 Sums
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§18.18(ii) Addition Theorems
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Ultraspherical
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Legendre
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Laguerre
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§18.18(iii) Multiplication Theorems
6: 1.9 Calculus of a Complex Variable
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DeMoivre’s Theorem
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Jordan Curve Theorem
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Cauchy’s Theorem
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Liouville’s Theorem
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Dominated Convergence Theorem
7: 12.13 Sums
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§12.13(i) Addition Theorems
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12.13.2 U ⁡ ( a , x + y ) = e 1 2 ⁢ x ⁢ y 1 4 ⁢ y 2 ⁢ m = 0 ( a 1 2 m ) ⁢ y m ⁢ U ⁡ ( a + m , x ) ,
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12.13.3 V ⁡ ( a , x + y ) = e 1 2 ⁢ x ⁢ y + 1 4 ⁢ y 2 ⁢ m = 0 ( a 1 2 m ) ⁢ y m ⁢ V ⁡ ( a m , x ) ,
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12.13.5 U ⁡ ( a , x ⁢ cos ⁡ t + y ⁢ sin ⁡ t ) = e 1 4 ⁢ ( x ⁢ sin ⁡ t y ⁢ cos ⁡ t ) 2 ⁢ m = 0 ( a 1 2 m ) ⁢ ( tan ⁡ t ) m ⁢ U ⁡ ( m + a , x ) ⁢ U ⁡ ( m 1 2 , y ) , ⁡ a 1 2 , 0 t 1 4 ⁢ π .
8: 24.4 Basic Properties
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24.4.12 B n ⁡ ( x + h ) = k = 0 n ( n k ) ⁢ B k ⁡ ( x ) ⁢ h n k ,
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24.4.13 E n ⁡ ( x + h ) = k = 0 n ( n k ) ⁢ E k ⁡ ( x ) ⁢ h n k ,
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24.4.14 E n 1 ⁡ ( x ) = 2 n ⁢ k = 0 n ( n k ) ⁢ ( 1 2 k ) ⁢ B k ⁢ x n k ,
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24.4.15 B 2 ⁢ n = 2 ⁢ n 2 2 ⁢ n ⁢ ( 2 2 ⁢ n 1 ) ⁢ k = 0 n 1 ( 2 ⁢ n 1 2 ⁢ k ) ⁢ E 2 ⁢ k ,
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Raabe’s Theorem
9: 1.4 Calculus of One Variable
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Mean Value Theorem
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Fundamental Theorem of Calculus
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First Mean Value Theorem
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Second Mean Value Theorem
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§1.4(vi) Taylor’s Theorem for Real Variables
10: 18.15 Asymptotic Approximations
►The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►
18.15.12 P n ⁡ ( cos ⁡ θ ) = ( 2 sin ⁡ θ ) 1 2 ⁢ m = 0 M 1 ( 1 2 m ) ⁢ ( m 1 2 n ) ⁢ cos ⁡ α n , m ( 2 ⁢ sin ⁡ θ ) m + O ⁡ ( 1 n M + 1 2 ) ,
►Another expansion follows from (18.15.10) by taking λ = 1 2 ; see Szegő (1975, Theorem 8.21.5). …