MIT OpenCourseWare
http://ocw.mit.edu
16.346 Astrodynamics
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 14 Hypergeometric Functions and Continued Fractions
John Wallis’ Hypergeometric Series
a + a(a + b) + a(a + b)(a + 2b) + · · · + a(a + b)(a + 2b) . . . [a + (n − 1)b] + · · ·
Hypergeometric Function
Named by Gauss’ mentor Johann Pfaff 1765–1825
In the year 1812, Carl Friedrich Gauss published his book entitled:
GENERAL INVESTIGATIONS
CONCERNING THE INFINITE SERIES
1+
αβ
α(α + 1)β(β + 1)
α(α + 1)(α + 2)β(β + 1)(β + 2) 3
x+
xx+
x +etc.
1·γ
1 · 2 · γ(γ + 1)
1 · 2 · 3 · γ(γ + 1)(γ + 2)
We use the symbol F (α, β; γ; x) to represent this series.
Examples of Hypergeometric Functions
3
x2 −
xF
α,
β
;
;
sin x = αlim
→∞
2 4αβ
β→∞
1
x2 −
cos x = αlim
F
α,
β;
;
→∞
2 4αβ
β→∞
log(1 + x) = xF (1, 1; 2; −x)
1
3
arctan x = xF , 1; ; −x2
2 2
x
ex = lim F α, 1; 1;
α→∞
α
Gauss’ Differential Equation
x(1 − x)
dy
d2 y
+
[γ
−
(α
+
β
+
1)x]
− αβy = 0
dx2
dx
has the general solution
y = c1 F (α, β; γ; x) + c2 x1−γ F (α − γ + 1, β − γ + 1; 2 − γ; x)
Gauss’ Continued Fraction Expansion
F0 = F (α, β; γ; x)
F1 = F (α, β + 1; γ + 1; x)
F1 − F0 = δ1 xF2
F2 = F (α + 1, β + 1; γ + 2; x)
F2 − F1 = δ2 xF3
F3 = F (α + 1, β + 2; γ + 3; x)
F3 − F2 = δ3 xF4
F4 = F (α + 2, β + 2; γ + 4; x)
F4 − F3 = δ4 xF5
16.346 Astrodynamics
Lecture 14
α(γ − β)
γ(γ + 1)
(β + 1)(γ − α + 1)
δ2 =
(γ + 1)(γ + 2)
(α + 1)(γ − β + 1)
δ3 =
(γ + 2)(γ + 3)
δ1 =
G0 =
F1
F0
G0 − 1 = δ1 xG1 G0
G0 =
1
1 − δ1 xG1
G1 =
F2
F1
G1 − 1 = δ2 xG2 G1
G1 =
1
1 − δ2 xG2
G2 =
F3
F2
G2 − 1 = δ3 xG3 G2
G2 =
1
1 − δ3 xG3
1
F (α, β + 1; γ + 1; x)
= G0 =
=
F (α, β; γ; x)
1 − δ1 xG1
1
=
δ1 x
1−
1−
1 − δ2 xG2
1
δ1 x
δ2 x
1−
1 − δ3 xG3
Since F (α, 0; γ; x) = 1, we have developed a continued fraction expansion for
F (α, 1; γ + 1; x)
Examples
log(1 + x) = xF (1, 1; 2; −x)
arctan x = xF ( 12 , 1; 32 ; −x2 )
arcsin x = xF ( 12 , 12 ; 32 ; x2 ) = x 1 − x2 F (1, 1; 32 ; x2 )
4
2ψ − sin 2ψ
= F (3, 1; 52 ; sin2 12 ψ)
Q=
3
3
sin ψ
arctanh x = xF ( 12 , 1; 32 ; x2 )
Sufficient Conditions for Convergence of Continued Fractions
Class I
Class II
a0
a0
a1
b0 +
a2
b1 +
b2 +
b2 −
a3
b3 − . . .
Will either converge or
diverge to infinity.
bn−1 bn
>0
n→∞
an
bn ≥ an + 1
lim
All an and bn are positive.
16.346 Astrodynamics
a2
b1 −
a3
b3 + . . .
Will either converge or
oscillate between two
different values.
Note:
a1
b0 −
Lecture 14
The Top-Down Method for Evaluating Continued Fractions
Pages 67–68
For a Class II continued fraction with n = 1, 2, . . . , we have
δn =
1−
1
an
bn−1 bn
where
δn−1
un = un−1 (δn − 1)
δ0 = 1
u0 = Σ0 =
Σn = Σn−1 + un
a0
b0
Continued Fractions Versus Power Series
For the tangent function
2
17 7
62 9
1
tan x = x + x3 + x5 +
x +
x + ··· =
3
15
315
2, 835
x
x2
1−
x2
3−
x2
5−
7 − ..
a. The series converges for − 12 π ≤ x ≤ 12 π .
b. The continued fraction converges for all x not equal to
16.346 Astrodynamics
Lecture 14
1
2
π ± nπ .
.