1

Chapter 11 Special functions

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 12 Gamma, beta, error, and elliptic

2

2. The factorial function (usually, n : integer)

1!!

32,

2Similarly,

.111

011

010

40

330

2

200

0

00

ndxexn

dxex

dxexdxex

dxexedxxe

edxe

xnn

xn

xx

xxax

xax

3

3. Definition of the gamma function: recursion relation (p: noninteger)

.0,0

1 pdxexp xp- Gamma function

ppp

ppdxexp xp

1

.1,!10

- Example .5/164/94/1,

4/14/14/54/5)4/5(4/9

so

!.1

,!1

0

0

1

ndxexn

ndxexn

xn

xn

- Recursion relation

4

4. The Gamma function of negative numbers

)0(11 ppp

p

- Example

.7.03.13.0

13.1,7.0

3.0

13.0

.0as11

. ppp

pcf

- Using the above relation, 1) Gamma(p= negative integers) infinite. 2) For p < 0, the sign changes alternatively in the intervals between negative integers

5

5. Some important formulas involving gamma functions

2/1

.442/1

.2211

2/1)prove(

2/

0 00 0

2

000

222

22

rdrdedxdye

dyeydyey

dtet

ryx

yyt

.sin

1p

pp

6

6. Beta functions

pqBqpBcfqpdxxxqpB qp ,,..0,0,1,1

0

11

yyxy

dyyqpBiii

xdqpBii

ayxdyyayaa

dy

a

y

a

yqpBi

qp

p

qp

a qpqp

aqp

1/.1

,)

sin.cossin2,)

/.1

1,)

0

1

212122/

0

0

1110

11

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7. Beta functions in terms of gamma functions

qp

qpqpB

,

.,2

1

2

1sincos4

sincos4

4

2,2

)Prove

0

2/

0

1212122

0

2/

0

1212

0 0

1212

0

12

0

12

0

1

2

2

22

22

qpBqpddrer

rdrderr

dxdyeyxqp

dxexqdyeydtetp

pqrqp

rpq

yxpq

xqyptp

8

- Example

0 5

3

1 x

dxxI

0

1

.1

,. qp

p

y

dyyqpBcf

.

4

1

!4

!3

5

14

.1,431,5

qppqp

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8. The simple pendulum

.sin0sin

cos2

1

cos

2

1

2

1

22

22

22

l

gmglml

dt

d

mglmlVTL

mglV

lmmvT

- Example 1 For small vibration,

./21

sin

glTl

g

10

- Example 2

integralellipticcf..constcos2

1

:sinsinsin

2

l

g

dl

gdor

l

g

l

g

In case of 180 swings (-90 to +90)

./42.7computer, Using

!function!Beta,cos2

4

.4

22

cos

.2

cos,cos

2,cos

2

1

.0const.090

2/

0

4/

0

2/

0

2

glT

d

g

lT

T

l

gdt

l

gd

dtl

gd

l

g

dt

d

l

g

T

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9. The error function (useful in probability theory)

.2

erf0

2

dtex t- Error function:

- Standard model or Gaussian cumulative distribution function

.2/erf2

1

2

1

2

1

2/erf2

1

2

1

2

1

2/

2/

2

2

xdtex

xdtex

x t

x t

.2

2erfc

,2/erf12

erfc

2/

2/

2

2

x

t

x

t

dtex

xdtex

- Complementary error function

.122erf xx

- in terms of the standard normal cumulative distribution function

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- Several useful facts

1.!253

2

!21

22erf

.12

12

2

1

2

122erf

erferf

53

0

42

0

0

2

2

xxx

xdtt

tdtex

dte

xx

xx t

t

- Imaginary error function:

xix

dtexx t

ierfierf

.2

erfi0

2

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10. Asymptotic series

.2

erf1erfc2

x

t dtexx

.1

2

1

2

1

1

2

1

2

11

2

11

,2

1111Using

22

2222

2222

2

2

x

tx

x

t

x

t

x

t

x

t

tttt

dtet

ex

dtt

eet

dtedt

d

tdte

edt

d

ttet

tet

e

14

.

1

2

3

2

13

2

1

2

11

2

1

//1/1Using

22222

22

4343

2132

x

tx

x

t

x

t

x

t

tt

dtet

ex

dtt

eet

dtedt

d

edtdtet

1.2

531

2

31

2

11~erf1erfc 32222

2

xxxxx

exx

x

- 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)

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11. Stirling’s formula

.2~288

1

12

1121

2~!

2pep

pppepp

nenn

pppp

nn

- Stirling’s formula

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11. Elliptic integrals and functions

- Legendre forms:

.10,sin1,:kindSecond -

,10,sin1

,:kindFirst-

0

22

0 22

kdkkE

kk

dkF

- Jacobi forms:

0 0 2

2222

0 0 22222

.1

1sin1,

,10,11sin1

,

sin,sin

dtt

tkdkkE

ktkt

dt

k

dkF

xt

x

x

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- Complete Elliptic integrals (=/2, x=sin=1):

2/

0

1

0 2

222

2/

0

1

0 2222

.1

1sin1,

2

,11sin1

,2

dtt

tkdkkEkEorE

tkt

dt

k

dkFkKorK

t

t

- Example 1

4/,3/sin,/1,2/3,

964951.0~21,3/,sin2/11

1

3/

0

22

EkEorkEkxEor

EkEd

18

- Example 2

3/

0

223/

0

2 sin2/114sin816

dd

.,2,

,2,.cf

,,sin1.cf 21222

1

kEnEknE

kFnKknF

kEkEdk

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- Example 4. Find arc length of an ellipse.

ellipseoftyeccentrici:,kindsecondtheofintegralelliptical

.sin1sin

.sincos

cos,sin

22

222

22

222222

22222222

ea

bak

da

baadbaads

dbadydxds

byax

(using computer or tables)

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- Example 5. Pendulum swing through large angles.

2sin24

2sin2

24

integralelliptic2

sin24

2

coscos

.coscos2

.constcos2

0

2

2

Kg

lK

g

lT

KT

l

gd

l

g

l

g

21

2/~2/sin,smallfor

1612

2sin

4

3

2

1

2sin

2

11

24

series.by ion approximat ,/2sinlargetoonotFor

24

22

2

212

g

l

g

lT

α

- For =30, this pendulum would get exactly out of phase with one of very small amplitude in about 32 periods.

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- Elliptic Functions

uuudu

d

xku

xu

uxxtkt

dtu

xt

dtu

x

x

dncnsn

1dn

1cn

.sn function) elliptic(sn11

sin1

22

2

1

0 222

1

0 2