常微分方程幂级数解法1

z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
dz
dt
d2w dz2
=
t4
d2w dt2
+
2t3 dw dt
d2w
dw
dz2
+ p(z) dz
+ q(z)w
=0
(1)
d2w dt2 + t=0 (1)
2 p(1/t)
− t
t2
(2)
()
dw q(1/t)
+ dt
t4
w=0
(2)
()
z=∞
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Formulation Ordinary Point & Singularity
1
2
3
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
d2w
wenku.baidu.comdw
dz2
+ p(z) dz
+ q(z)w
=0
p(z) q(z)
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
=0
γ − (1 + α + β)z p(z) =
z(1 − z)
αβ q(z) = −
z(1 − z)
z=0 z=1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
z0
p(z), q(z) z0
Laurent
z0 Taylor
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
− t
t2
+ dt
t4
w=0
p(z)
=
2 z
+
a2 z2
+
a3 z3
+
Analytic Continuation
Formulation Ordinary Point & Singularity
p(1/t) = 2t + a2t2 + a3t3 + · · · q(1/t) = b4t4 + b5t5 + · · ·
t=0
d2w 2 p(1/t) dw q(1/t)
dt2 +
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2
l(l + 1) q(z) = 1 − z2
z = ±1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.2
(hypergeometric)
d2w
dw
z(1 − z) dz2
+ [γ − (1 + α + β)z] dz
− αβw
=0
γ − (1 + α + β)z p(z) =
z(1 − z)
αβ q(z) = −
z(1 − z)
z=0 z=1
d2w
dw
dz2
+ p(z) dz
+ q(z)w
=0
p(z) q(z)
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
z=∞
z = 1/t
dw = −t2 dw
z0
p(z), q(z)
z0
z0
9.2
(hypergeometric)
d2w
dw
z(1 − z) dz2
+ [γ − (1 + α + β)z] dz
− αβw
=0
γ − (1 + α + β)z p(z) =
z(1 − z)
αβ q(z) = −
z(1 − z)
z=0 z=1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.2
(hypergeometric)
d2w
dw
z(1 − z) dz2
+ [γ − (1 + α + β)z] dz
− αβw
Formulation Ordinary Point & Singularity
z=∞
z = 1/t
dw = −t2 dw
dz
dt
d2w dz2
=
t4
d2w dt2
+
2t3 dw dt
d2w
dw
dz2
+ p(z) dz
+ q(z)w
=0
(1)
d2w dt2 + t=0 (1)
2 p(1/t)
− t
t2
Analytic Continuation
References
§6.1 — 6.2
§9.1, 9.2
§8.1
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
1
2
3
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2
Outline
()
2007
C. S. Wu
()
Outline
1 2
3
C. S. Wu
()
Outline
1 2
3
C. S. Wu
()
Outline
1 2
3
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
(2)
()
dw q(1/t)
+ dt
t4
w=0
(2)
()
z=∞
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
z0
p(z), q(z) z0
Laurent
z0 Taylor
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
d2w
dw
dz2
+ p(z) dz
+ q(z)w
=0
p(z) q(z)
C. S. Wu
()
ODE: Ordinary Point & Singularity Solutions in Vicinity of Ordinary Point
Analytic Continuation
Formulation Ordinary Point & Singularity
z=∞
z = 1/t
dw = −t2 dw
dz
dt
d2w dz2
=
t4
d2w dt2
+
2t3 dw dt
d2w
dw
dz2
+ p(z) dz
+ q(z)w
=0
(1)
d2w dt2 + t=0 (1)
2 p(1/t)
− t
t2
(2)
()
dw q(1/t)
+ dt
t4
w=0
(2)
()
z=∞
C. S. Wu
Analytic Continuation
Formulation Ordinary Point & Singularity
p(z), q(z) z0
z0
p(z), q(z)
z0
z0
9.1 Legendre
(1
−
z2)
d2w dz2
−
dw 2z
dz
+
l(l
+
1)w
=
0
2z p(z) = −1 − z2