Transcript chapter 11 Special functions
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 11 Special functions
Lecture 12 Gamma, beta, error, and elliptic 1
2. The factorial function (usually, n : integer) 0
e ax dx
1
e
x
0 1 0
xe
ax dx
1
xe
x
0 0 Similarly, 0
x
2
e
x dx
2 3 ,
0
1
e
x dx
1 2 .
0
x
3
e
x dx
2 3 4 0
x n e
x dx
n
!
n
1 0
x n e
x dx
n
!
1
2
3. Definition of the gamma function: recursion relation ( p : noninteger) - Gamma function 0
x p
1
e
x dx
,
p
0 .
0
x n
1
e
x dx
n
1
!
,
n
1
0
x n e
x dx
n
!.
- Recursion relation
p
1
0
x p e
x dx
p
1
p
p
!
,
p
1 .
- Example
so
, 1 / 4 ( 5 / 4 ) 9 /
16 / 5 .
3
4. The Gamma function of negative numbers
1
p
p
1
(
p
0 ) - Example
0 .
3
1 0 .
3
,
1 .
3
.
cf
.
1
p
p
1
as
p
0 .
- Using the above relation, 1) Gamma( p = negative integers) infinite.
2) For p < 0, the sign changes alternatively in the intervals between negative integers 4
5. Some important formulas involving gamma functions ( prove ) 0 1
t e
t dt
0 1
y e
y
2 2
ydy
2 0
e y
2
dy
.
2 4 0 0
e
x
2
y
2
dxdy
4 0 / 2 0
e r
2
rdrd
.
1
p
sin
p
.
5
6. Beta functions
B p
, 1 0
x p
1
1
x
q
1
dx
,
p
0 ,
q
0 .
cf
.
B p
,
q
,
i
)
ii
)
B
0
a y a B
2 0 / 2
p
1 1
sin 2
a y q
1
dy a
a
1
p
q
1 0
a y p
1
a
y
q
1
dy
.
1 cos
2
q
1
d
.
x
sin 2
x
y
/
a
iii
)
B
, 0
1
y
p
1
dy y
p
q
.
x
y
/
1
y
6
7. Beta functions in terms of gamma functions
B
p
q q
Prove ) 0
t p
1
e
t dt
2 0
y
2
p
1
e
y
2
dy
, 4 0 0
x
2
q
1
y
2
p
1
e
x
2
y
2
dxdy
4 0 0 / 2
r
cos
r
sin
2
p
1
e
r
2
rdrd
2 0
x
2
q
1
e
x
2
dx
4 0
r
2
p
2
q
1
e
r
2
dr
0 / 2
cos
sin
2
p
1
d
1 2
p
q
1 2
B
.
7
- Example
I
0
1
x
3
dx x
5
cf
.
B
0
1
y
p
1
dy y
p
q
.
p
q
5 ,
1
p
1 3 3 !
4 !
1 4 .
p
4 ,
q
1 .
8
8. The simple pendulum
T
1 2
mv
2
V
1 2
m l
2
mgl
cos
L
T
V
1 2
ml
2 2
mgl
cos
d dt
ml
2 2
mgl
sin 0
g l
sin .
- Example 1 For small vibration, sin
l g
T
1 2
l
/
g
.
9
- Example 2
l g
sin
l g
sin 1 2 2
l g
cos const .
cf.
elliptic
or
d
integral
l g
sin
d
: In case of 180
90
0 swings (-90 const.
0 .
to +90 ) 0 1 2 2
l g
cos , / 2
d
dt
2
g l
cos ,
d
cos 2
g dt
.
l d
cos
T
4 2
g l
0
T
/ 4
dt
l
2
g
0 / 2 2
g l
T
4 .
d
cos , Beta function!
!
Using computer,
T
7 .
42
l
/
g
.
10
9. The error function (useful in probability theory) - Error function: erf 2 0
e
t
2
dt
.
- Standard model or Gaussian cumulative distribution function 1 2
x
e
t
2 / 2
dt
1 2 1 2 erf
x
/ 2 1 2 1 2
x
e
t
2 / 2
dt
1 2 erf
x
/ 2 .
- Complementary error function erfc 2
x
e t
2 / 2
dt
1 erf
x
/ 2 , erfc
x
2 2
x
e t
2 / 2
dt
.
- in terms of the standard normal cumulative distribution function erf
2 1 .
11
- Several useful facts erf erf erf erf 2 0
e t
2
dt
2 0
x e
t
2
dt
2 1 2 1 2 2 1 2 1 .
2 0
x
1
t
2
t
4 2 !
dt
2
x
x
3 3
x
5 5 2 !
.
x
1 - Imaginary error function: erfi erf 2 ierfi 0
x e t
2
dt
.
12
10. Asymptotic series erfc 1 erf 2
x
e t
2
dt
.
Using
e
t
2
t
1
te
t
2
t
1
te
t
2 1
t d dt
x
e t
2
dt
x
1
t d dt
1 2
e
t
2
dt
1
t
1 2
x e
x
2 1 2
x
t
2 1
e
t
2
dt
.
1 2
e
t
2 , 1 2
e
t
2
x
x
1 2
e
t
2 1
t
2
dt
13
Using
x
d dt
e
t
2
d
/
dt
1 2
e
t
2 1 2
e
t
2
dt
1
t
3 1 2
e
t
2
x
x
1 2
e
t
2 3
t
4
dt
1 2
x
3
e
x
2 3 2
x
t
4 1
e
t
2
dt
.
erfc 1 erf ~
x e
x
2 1 1 2
x
2 2 1
x
2 3 2 1 3
x
5 3 .
x
1 - This series diverges for every x because of the factors in the numerator. For large enough x , the higher terms are fairly small and then negligible. For this reason, the first few terms give a good approximation. (asymptotic series) 14
11. Stirling’s formula - Stirling’s formula
n
!
~
n n e
n
2
n
p
1
p p e
p
2
p
1 1 12
p
1 288
p
2 ~
p p e
p
2
p
.
15
11. Elliptic integrals and functions - Legendre forms: First kind :
F
Second kind :
E
0 0
d
1
k
2 sin 2 , 0
k
1 , 1
k
2 sin 2
d
, 0
k
1 .
- Jacobi forms:
t
sin ,
F
0
x
sin
d
1
k
2 sin 2 0
x
1
t
2
dt
1
k
2
t
2 ,
E
0 1
k
2 sin 2
d
0
x
1
k
2
t
2
dt
.
1
t
2 0
k
1 , 16
- Complete Elliptic integrals ( = /2, x =sin =1):
K or K E or E
F
2 ,
k
0 / 2
d
1
k
2 sin 2 1 0 1
t
2
dt
1
k
2
t t
,
E
2 ,
k
0 / 2 1
k
2 sin 2
d
1 0 1
k
2
t t dt
.
1
t
2 - Example 1 0
or
/ 3
E
1
2 sin 2
d
E
3 / 2 , 1 /
E k
or k
E
/ 3 , 1
E
, sin 1
k
2 ~ 0 .
964951
E
/ 3 , / 4
17
- Example 2 0 / 3 16 8 sin 2
d
4 0 / 3 1
2 sin 2
d
cf .
1 2 1
k
2 sin 2
d
E
1 ,
k
cf .
F
n
E
n
,
k
,
k
2
nK
2
nE
F
k E
.
2 ,
k
18
- Example 4. Find arc length of an ellipse.
x
ds
2
a
sin ,
dx
2
y dy
2
b
a
cos 2 cos 2
b
2 sin 2
d
2 .
ds
elliptical
a
2
a
2
b
2
sin 2
d
a
integral of the second 1
a
2
a
2
b
2 sin 2
d
.
kind ,
k
2
a
2
a
2
b
2
e
2 : eccentrici ty of ellipse (using computer or tables) 19
- Example 5. Pendulum swing through large angles.
2 2
g
cos
l
const .
2 2
g
cos
l
cos
.
0 cos
d
cos 2
g l T
4
T
4
l
2
g
2
K
sin 2 4
l g
2
K
sin 2 2
K
sin 2
elliptic integral
20
For
α T
4 not too large sin 2 1 2 , approximat ion by series.
l g
2 2 1 1 2 sin
for small 2 2 , 1 2 3 4 2 sin 4 2 sin / 2 ~ / 2
2
l g
1 2 16 - For =30 , this pendulum would get exactly out of phase with one of very small amplitude in about 32 periods.
21
- Elliptic Functions
u
0
x dt
1
t
2 sin 1
x u
0
x
1
t
2
dt
1
k
2
t
2 sn 1
x
( elliptic function)
x
sn
.
cn 1
x
2 dn
d du
sn 1
k
2
x
2 cn 22