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11: 12.2 Differential Equations
§12.2 Differential Equations
§12.2(i) Introduction
§12.2(iii) Wronskians
§12.2(iv) Reflection Formulas
§12.2(v) Connection Formulas
12: 12.3 Graphics
§12.3(i) Real Variables
See accompanying text
Figure 12.3.8: V ( a , x ) , - 2.5 a 2.5 , - 2.5 x 2.5 . Magnify 3D Help
§12.3(ii) Complex Variables
13: 12.12 Integrals
§12.12 Integrals
Nicholson-type Integral
12.12.4 ( U ( a , z ) ) 2 + ( U ¯ ( a , z ) ) 2 = 2 3 2 π Γ ( 1 2 - a ) 0 e 2 a t + 1 2 z 2 tanh t sinh ( 2 t ) d t , a < 1 2 .
See also Barr (1968) and Lowdon (1970).
14: 12.8 Recurrence Relations and Derivatives
§12.8(i) Recurrence Relations
12.8.2 U ( a , z ) + 1 2 z U ( a , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.3 U ( a , z ) - 1 2 z U ( a , z ) + U ( a - 1 , z ) = 0 ,
12.8.6 V ( a , z ) - 1 2 z V ( a , z ) - ( a - 1 2 ) V ( a - 1 , z ) = 0 ,
§12.8(ii) Derivatives
15: 12.13 Sums
§12.13 Sums
§12.13(i) Addition Theorems
12.13.1 U ( a , x + y ) = e 1 2 x y + 1 4 y 2 m = 0 ( - y ) m m ! U ( a - m , x ) ,
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 ) ,
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 ) ,
16: 10.39 Relations to Other Functions
Parabolic Cylinder Functions
10.39.4 K 3 4 ( z ) = 1 2 π 1 2 z - 3 4 ( 1 2 U ( 1 , 2 z 1 2 ) + U ( - 1 , 2 z 1 2 ) ) .
17: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
12.9.1 U ( a , z ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π - δ ( < 3 4 π ) ,
12.9.2 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s , | ph z | 1 4 π - δ ( < 1 4 π ) .
12.9.4 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s ± i Γ ( 1 2 - a ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , - 1 4 π + δ ± ph z 3 4 π - δ .
§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
18: 12.5 Integral Representations
§12.5(i) Integrals Along the Real Line
12.5.1 U ( a , z ) = e - 1 4 z 2 Γ ( 1 2 + a ) 0 t a - 1 2 e - 1 2 t 2 - z t d t , a > - 1 2 ,
§12.5(ii) Contour Integrals
§12.5(iii) Mellin–Barnes Integrals
§12.5(iv) Compendia
19: 29.16 Asymptotic Expansions
The approximating functions are exponential, trigonometric, and parabolic cylinder functions.
20: 12.4 Power-Series Expansions
§12.4 Power-Series Expansions
12.4.1 U ( a , z ) = U ( a , 0 ) u 1 ( a , z ) + U ( a , 0 ) u 2 ( a , z ) ,
12.4.2 V ( a , z ) = V ( a , 0 ) u 1 ( a , z ) + V ( a , 0 ) u 2 ( a , z ) ,