УрМатФиз / УрМатФиз с теорией

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Национальный исследовательский университет «МЭИ»

Предмет:

Высшая математика

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!k ' d' = Ak + Bk

.pdf

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	= 0	n	1	2
n		k
n		k			n	=	0 1 k = 1 1
k	B	a	C		n		0 1 k = 1 1
	@		A

n | k- n- :

Jn n = 0:

2:16

2:17

!" # $ % & ' # ( 2.15 :

0 n r1

Rn r = Jn B k C n = 0 1 k = 1 1

@ a A

Rn r 2 = Z

R2 r r dr = a2 J0 n 2:

n n k

+, # $,( #, $ % & , , .,& , , , ' %# # 1- , 2.2 # !

vnk r ' = Jn B

C An cos n' + Bn sin n'

n =

0 1

k =

1 1

jAnj + jBnj 6= 0

a 28An Bn

2:18

jjvnkjj

= Z

Z vnk r ' r drd' = jj0n ' jj

jjRn r jj

, $ % ( ,' ,2 & # 2.16 ,

, 2.17 .

$ 3 & Rn r & ' # (, ,' 4 #,|. 35 2.12 #

Rn a = 0
0
jRn0j +1:	2:19
jRn0j +1:

7, &. , , , ' %# & r = 0 8 # 35 2.12

! 2.15
	Rn r = Jn p		:

, # 2	, ' 2.19 & ' # &
, & $ %2 ( ,'
	Jn0 p	= 0:

9 ! , & ', # $ % ( ,' :

331

	= 0	n	1	2
n		k
n		k			n	=	0 1 k = 1 1	2:20
k	B	a	C		n		0 1 k = 1 1	2:20
	@		A

n | k- n- :

			0	n	= 0:												2:21
		Jn k			= 0:												2:21
	! " # $% ! & 2.15 :
	0	k	r1
		n
Rn r = Jn B		a C n				=		0 1		k =				1 1
	@		A
2	a				a2	0			0	n	1	2	1		2	n
2	2					0			0		1		1		2
R n r = Z Rn r r dr =							1
R n r = Z Rn r r dr =					2	B	1		@	n	A		C		Jn k		:
	0				2	B				k			C
	0					@							A

)* ! " *& !, " # $* * ,*$* * * % #! ! 2- * 2.3 !

0		k	r1
		n
vnk r ' = Jn B		a	C An cos n' + Bn sin n'			n =		0 1	k
@			A		jAnj + jBnj 6= 0
			a 28An Bn		jAnj + jBnj 6= 0
jjvnkjj	2	= Z Z		2		2	2
jjvnkjj		= Z Z		vnk r ' r drd' = jj/n ' jj			jjRn r jj
		0	0

=1 1

2:22

n	\|
* " # & % 0 $ ! 2.20 , k	\|
* 2.21 .

1 $ Rn r $% ! &* *% 2 !*|, 13 2.12 !

Rn + hRn = 0
0
jRn0j +1:	2:23
jRn0j +1:

4 * % #! $ r = 0 5 ! 13 2.12 -

2.15	Rn r = Jn p :

* ! 0 * % # 2.23 $% ! $
* $ " #0 & *% :
p	n0 p	+ hJn0 p	= 0:

4 * $%* ! " # & *% :

332

= 0

0 1

k = 1 1

2:24

| k-

2:25

Jn + haJn = 0:

! " #$ % & '

( 2.15 :

0 k

Rn r = Jn B

a C n

0 1

1 1

0 n 121 2

n 3

a2 2 0 n

R n r = Z Rn r r dr =

Jn k

! (

, ! " #$ % -%&

' "

3- 2.4

0 k

vnk r ' = Jn B

C An cos n' + Bn sin n'

n =

0 1

k =

1 1

A8An Bn jAnj + jBnj 6= 0

a 2

jjvnkjj

= Z

Z vnk r ' r drd' = jj/n ' jj

jjRn r jj

! " ( ' 0 % # & 2.24 ,

2.25 .

. 1 " ! " #$ % ( & #$ 2 2.22 0 "0 ( ' & , % "0 0 3 4 0 ( ' % #-

& 2.16 & 2.19 .

333

D : r a 0

0 ' 2

4v r ' + v =

@2v

3:1

1 @

2 @v

! +

sin

! +

+ v = 0

r2 sin

r2 sin2

@'2

	v	a	= 0
		a
@v@r			ra = 0

@r + hv r a= 0:

3:2

3:3

3:4

.			!-
#	v r ' = R r Y ' 6	0:	3:5

3.5 3.1 ! :

(

r2 dRdr ! + r2R

4' Y '

R r

Y '

@2Y

4' Y =

sin

! +

sin

sin2

@'2

R r ! + ,*-

R r + 2rR r + r2 ! R r = 0

3:6

3:7

Y ' + 2 -! + ! ' ( +-! = 0 = | + 0 |1 :

1			@		@Y		1		@2Y
4' Y + Y =				sin	@ ! +					+ Y
	sin	@				sin2			@'2
										3:8
0 0 ' 2
jY 0j			+1		jY j			+1		3:9

334

Y = Y			@Y = @Y		:	:10
			@'	@'
			-
	Y	' = ! ' Z :				:11
	Y	' = ! ' Z :				:11

"			#
1		d	dZ	! + Z
			sin d
	sin	d	sin d

Z = sin2

=! ' = &:

' (# )* Z " # #" + , " = 0= # #

1	d	dZ	&
		sin d	! +		! Z = 0	2	:12
sin	d	sin d	! +	sin2	! Z = 0	2	:12
			jZ j +1 jZ j +1:				:13

' (# )* ! ' " # #" + " ' #

# -# \|. # :
!	' + &! ' = 0			' 2			:14
	! = !		!	= !	:		:15
			#		. 1		. . 2 3"-

	&k = k2		k =				:16
	&k = k2		k =	0 1			:16
" " #5 " 3" (# )*							5
" #5 + :
!k ' = Ak cos k' + Bk sin k'					k =
!k ' = Ak cos k' + Bk sin k'					k =	0 1
	8An Bn	jAnj + jBnj 6= 0					:17

jj!0jj	2	2	2	jj!kjj	2	=
		= A0

" & = &k					7'8			" # -
" + + x = cos , Z = Z								x y x : -
"				#			# # -# \|
. # 7'8 " (# )* + . 9 :
	d	x2 dy	+	0	k2	1 y x = 0 1 x 1			:18
	dx				2
		dx!		@	1 x	A

335

jy 1j +1:

:19

n = n n + 1 n =

:20

0 1

! "# | -

yx = Pnk x x2

k=2

Pn x n =

0 1 k = 0 n&

dxk

jjPnk jj2 = Z1

n + k!

Pnk x 2dx =

+ 1

n k!

( Pnx |

, "

( * +

! , - (

Pn x =

x2 1n n = 0 1:

2nn!

dxn

. " + ,* +

+ , -

! "#

Znk = Pnk

n =

k =

:21

0 1

1 ",

! "# 2

|3 ,,

, ! "#

Ynk ' = Pnk

Ak cos k' + Bk sin k'

:22

n = 0 1 k = 1 1

+ , +

= n 56

+ ,

n n + 11

R r + rR r + @

A R r = 0 n = 0 1:

3:23

R r = y r =p

" # #$# %

n + 1=22 1

y + ry

A y = 0

& ' %

y r = CnJn+1=2 p r + DnNn+1=2 p

r :

( # % )

jR0j +1:

3:24

!) % ) +$, -, Nn+1=2 p

r r ! 0

# % Dn = 0 # ) % ) r ! 0 '

3.23

336

R r =	Jn+1=2 p		r	:	3:25

		pr
3.2 \| 3.4 .
3.2 ,
R a = 0:					3:26
$% &		' \|( 3.23 ,

3.24 , 3.26 .

3.26 3.25

Jn+1=2 p a = 0:

3:27

* + , &

nm = 0

n+1=2

m a

n =

m =

3:28

0 1

1 1

mn+1=2 | m-% 0

Jn+1=2 = 0

+1 & 2 03 3.27 3.28 +

Rnm r = Jn+1=2 0

n+1=2

r1, p

m a

m =

3:29

1 1

n+1=2

r1 r dr =

jjRnmjj2 = Z Rnm2 r2 dr = Z Jn2+1=2 0 m a

Jn+1=2 mn+1=2

3:30

0, & 2 03 5 + 3.5 , 3.22 , 3.29

vnkm r ' = Ynk ' Rnm r =

n+1=2

r1, p

= Ynk ' Jn+1=2

0 m a

m =

3:31

1 1

+ 2-

+1 & nm

3.28 . $ & % 2 03 3.17 , 3.20

.30 jjvnkmjj2 = jj6kjj2 jjPnk jj2 jjRnmjj2: 3:32

337

3.3 ,

a = 0

, -

3.25 , "

a Jn+1=2 p

2p aJn+1=2 p

a = 0:

#$ "

nm = 0

n+1=2

m a

n =

0 1

1 1

mn+1=2 | m- ' (

2 Jn+1=2 Jn+1=2 = 0

$) " * '+ "$

n+1=2

Rnm r = Jn+1=2 0 m a

r1, p

n =

m =

3:33

0 1

1 1

@ a

n+1=2

1 r dr =

jjRnmjj2 = Z

Rnm2 r2 dr = Z Jn2+1=2 0

2 Jn+1=2 mn+1=2

n + 1=2

1 Jn2+1=2 mn+1=2 3 :

3:34

n+1=2

3.4 ,

R a + hR a = 0

, -
3.25 , "
2p aJn+1=2 p	a + 2ah 1Jn+1=2 p					a = 0:
2p aJn+1=2 p	a + 2ah 1Jn+1=2 p					a = 0:
#$ "
n+1=2			2
nm = 0 m a		1
nm = 0 m a		1	n =	0 1	m =		1 1
@		A

mn+1=2 | m- ' (

2 Jn+1=2 + 2ah 1Jn+1=2 = 0

$) " * '+ "$

Rnm r = Jn+1=2 0		n+1=2	r1, p
		m a			n =		m =
				r
						0 1		1 1
" 3.34 .	@		A

338

	1	2
	r + r R r	r2 R r = 0 0:

= 0

R0 r + 1rRr r = 0:

, , y r = R0 r:

y r + 1ry r = 0 , dyyr = drr , y r = D0 1r:

! " #

dR0	= B0	1	, R0 r = C0 + D0 ln r:
dr		r

4:1

4:2

6=0 % # " & ' 4.1
R r = r :	4:3
" 4.3 4.1 #
r2 1 + 2 = 0 , 2 = 2 , 1 2 = :

* + , % & % " &, # " &, '-| r r . !+ ' % / ! 4.1 6= 0 ' " % % + 0 1 , # " &, ' %:

6= 0

+ Rn00 r +

2Rn r n n + 1Rn r = 0

n =

4:5

0 1

3 % # " & ' 4.5 4.3 . " 4.34.5 #

r 2 1 + 2 n n + 1 = 0 , 2 + n n + 1 = 0 ,

, 1 = n 2 = n + 1:

* + , % & % " &, # " &, '-| rn r n+1 . !+ ' % / ! 4.5 " " + % % + 0 1 , # " &, ' %:

Rn r = Cnrn + Dnr n+1 n =		:	4:6
	0 1

339

2	0	2	n n + 11

n r + rRn r + @k			r2 A Rn r = 0 n = 0 1:	4:7

x = kr:

Rn r = Rn x=k

R x :

dR x

dR dx

d2R x

= dx dr

= R x k

dr2

= R x k

# 4.7

n n + 11

R x + xR x + @1

A R x = 0:

% :

R x =

y x

y" x

y x

R x =

x 2

2x3 2

R x =

x1 2

x3 2

x5 2

( )* ) + ,# - . (- 0 y x ,# 1- 0 -( :

1 0	n + 1=22 1
y" + xy + @1	x2 A y = 0:

2 34 5 + %

y x = AnJn+1=2 x + BnNn+1=2 x

Jn+1=2 x, Nn+1=2 x | . (0 1- 8 0 )* --( ,

y x = CnHn1+1=2 x + DnHn2+1=2 x

Hn1+1=2 x = Jn+1=2 x + iNn+1=2 x Hn2+1=2 x = Jn+1=2 x iNn+1=2 x

| . (0 9 (- 1- 2- 0 -( .

( 3 , 34 5 ,# 4.7 + %

Jn+1=2 kr

Nn+1=2 kr

Rn r = An

+ Bn

n = 0 1

Hn1+1=2 kr

Hn2+1=2 kr

+ Dn

n = 0 1:

Rn r = Cn

4:8

4:9

340

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