7/23/2019 Mathematical Physics Notes

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I. Infinite Seriesa. Power Series

McLaurin Series

or f(x)=n=0

f(n )(0)i!

xn

b. Fourier Series

f(x)=ao+ann=1

cos nx+bnn=1

sin nx

p=2L

ao=1

2LL

L

f(x)dx

an=1

LL

L

f(x)cosnx L

dx

bn=1

L L

L

f(x)sinnx L dx

c. Complex Series

einx=cosnx+ isin nx

einx=cos nxi sinnx

cosnx=e

inx+einx

2

sinnx=einxeinx

2 i

an cosnx+bnsin nx=1

2(anibn)e

inx+1

2(an+ ibn)e

inx

c n=an+i bn

c n*=ani bn

f(x)=co+n=1

cn einx+

n=-

1

c n einx=

n=

cn einx

when p=2L

Cn=1

2LL

L

f(x)e(i nx)/L

II. Special Functionsa. amma Function

(x)=0

tx1

e t

dt

(1/2)=(n)=(n1)(n1)

b. Stirlin!"s Formula #for lar!e x$x!=(x+1)2 e

xx

x1/2

c. %eta Function

B (p , q)=(p)+(q )(p+q)