About the Project

classification%20of%20parameters

AdvancedHelp

(0.003 seconds)

1—10 of 446 matching pages

1: 20 Theta Functions
Chapter 20 Theta Functions
2: 31.14 General Fuchsian Equation
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z a j ) d w d z + ( j = 1 N q j z a j ) w = 0 , j = 1 N q j = 0 .
α β = j = 1 N a j q j .
The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N 2 free accessory parameters q j . With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z a j ) γ j / 2 ) W ( z ) ,
3: 16.21 Differential Equation
16.21.1 ( ( 1 ) p m n z ( ϑ a 1 + 1 ) ( ϑ a p + 1 ) ( ϑ b 1 ) ( ϑ b q ) ) w = 0 ,
With the classification of §16.8(i), when p < q the only singularities of (16.21.1) are a regular singularity at z = 0 and an irregular singularity at z = . …
4: 31.2 Differential Equations
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, { } , can be transformed into (31.2.1). The parameters play different roles: a is the singularity parameter; α , β , γ , δ , ϵ are exponent parameters; q is the accessory parameter. The total number of free parameters is six. … Then (suppressing the parameter k ) … Except for the identity automorphism, each alters the parameters.
5: 31.12 Confluent Forms of Heun’s Equation
31.12.1 d 2 w d z 2 + ( γ z + δ z 1 + ϵ ) d w d z + α z q z ( z 1 ) w = 0 .
31.12.2 d 2 w d z 2 + ( δ z 2 + γ z + 1 ) d w d z + α z q z 2 w = 0 .
31.12.3 d 2 w d z 2 ( γ z + δ + z ) d w d z + α z q z w = 0 .
31.12.4 d 2 w d z 2 + ( γ + z ) z d w d z + ( α z q ) w = 0 .
6: 8 Incomplete Gamma and Related
Functions
7: 28 Mathieu Functions and Hill’s Equation
8: 16.8 Differential Equations
§16.8(i) Classification of Singularities
16.8.3 ( ϑ ( ϑ + b 1 1 ) ( ϑ + b q 1 ) z ( ϑ + a 1 ) ( ϑ + a p ) ) w = 0 .
9: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 10: 23 Weierstrass Elliptic and Modular
    Functions