THE ASSOCIATED LEGENDRE FUNCTIONS: m dm P" (x) = (1 – x²) dxm P(x) Example: Make the change of variable x= cos 8 in Sing do Si79 To find:
THE ASSOCIATED LEGENDRE FUNCTIONS: m dm P" (x) = (1 – x²) dxm P(x) Example: Make the change of variable x= cos 8 in Sing do Si79 To find:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Explain this proof in very boring details please
Transcribed Image Text:THE ASSOCIATED LEGENDRE FUNCTIONS:
dm
P.(x)
m
P™ (x) = (1 – x²).
dxm
Example:
Make the change of variable x= cos 8 in
m
To find:
Si79
(1 – x²) y" – 2xy' +
m2
- 1) –
y = 0
with
m² s1?
1- x2
Somtion:-
dz
cos8 >
sind
00
dy
dy
dy
(- sin@)
%3D
dy
- sing
dy
oy sin'g + cos'8 = 1 → sin'9 = Vx²
%3D
brom (2) into c1):
dy
do
dy
%3D
|
- VT-x²
%3D
doc
dy
%3D
%3D
de de
d'y
dx
= - - z'
d'y
![dy
%D
xp
OP
For a given eguation ;-
(Sin 8 dy
]y= 0
Sinig
Sind dg
we get :
(Sin8
d, ]y= o
cos 8 d)+[ eceai) -
Sin'g
Sin e
cos 8 dy +[ec&+1) -
Sin g do
Sin? g
From (3) and (5)
into (6)
d'y
dy
dy
+ [ ecepi) - ]y = 0
Jy =0
|
d'y
dy
dy
ece t)-
y] = 0
%3D
(1- x²)
dy
dy
- 2 x
+ [ ec& +1) - Jy=0
dx?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7016d9cb-ec91-404b-ab58-196ae7860125%2F655bda59-f071-4f3c-a2e6-32f0fa2b9fac%2Fvqkbgxza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:dy
%D
xp
OP
For a given eguation ;-
(Sin 8 dy
]y= 0
Sinig
Sind dg
we get :
(Sin8
d, ]y= o
cos 8 d)+[ eceai) -
Sin'g
Sin e
cos 8 dy +[ec&+1) -
Sin g do
Sin? g
From (3) and (5)
into (6)
d'y
dy
dy
+ [ ecepi) - ]y = 0
Jy =0
|
d'y
dy
dy
ece t)-
y] = 0
%3D
(1- x²)
dy
dy
- 2 x
+ [ ec& +1) - Jy=0
dx?
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