Download Notes on Math - General Notes and more Study notes Mathematics in PDF only on Docsity!
Mathematical Formulae
1. Vector Formulae
Bold characters are vector functions and f is a scalar function.
A (B C) = C (A B) = B (C A) A (B C) = B(A C) C(A B) (A B) (C D) = (A C)(B D) (A D)(B C) r (f A) = rf A + f r A r (f A) = rf A + f r A r(A B) = A (r B) + B (r A) + (B r)A + (A r)B r (A B) = B (r A) A (r B) r (A B) = Ar B Br A + (B r)A (A r)B r rf 0 r (r A) 0 r (r A) = r(r A) r^2 A r r = 3; r = position vector r r = 0
r^2
jr r^0 j =^ ^4 (r^ ^ r
Substantive derivative: df dt
= @f @t
Substantive derivative: dA dt
= @A
@t
Substantive derivative: dv dt
= @v @t
+^1
rv^2 + (r v) v
Gaussítheorem:
Z
V
r AdV =
I
S
A dS
Stokesítheorem
Z
S
r A dS =
I
C
A dl I
S
(r A) dS = 0 for closed surface Z
V
r AdV =
I
S
dS A =
I
S
A dS
2. Delta Function
Z
f (t)(t t^0 )dt = f (t^0 )
(at) = 1 jaj
(t);
Z
g(t)[f (t)]dt = g(t
df dt (^) t=t 0
where f (t^0 ) = 0
(r r^0 ) (3-D delta function) = (x x^0 )(y y^0 )(z z^0 ) (Cartesian)
=
(r r^0 ) rr^0
( ^0 )
sin ( ^ ^
(^0) ) (spherical)
= (r^ ^ r
rr^0 (cos^ ^ ^ cos^
(^0) )( (^0) ) (spherical)
= (^ ^
( ^0 )(z z^0 ) (cylindrical)
3. Curvilinear Coordinates
Let ui(x; y; z) (i = 1; 2 ; 3) be a system of curvilinear coordinates. The metric coe¢ cients are
hi =
s @x @ui
@y @ui
@z @ui
and the length segments in each direction are hiduiei (ei unit vector). Area elements dSi = hj hkduj dukej ek Volume element dV = h 1 h 2 h 3 du 1 du 2 du 3 Gradient rf =
X^3
i=
hi
@f @ui Divergence
r A = 1 h 1 h 2 h 3
@u 1
(h 2 h 3 A 1 ) + @ @u 2
(h 3 h 1 A 2 ) + @ @u 3
(h 1 h 2 A 3 )
Curl
r A =
e 1 h 2 h 3
e 2 h 3 h 1
e 3 h 1 h 2 @ @u 1
@u 2
@u 3
h 1 A 1 h 2 A 2 h 3 A 3 Scalar Laplacian
r^2 f = r rf = (^) h^1 1 h 2 h 3
@u 1
h 2 h 3 h 1
@f @u 1
h 3 h 1 h 2
@f @u 2
h 1 h 2 h 3
@f @u 3
Vector Laplacian
r^2 A r(r A) r (r A)
=
r^2 Ar
r^2 Ar^ ^
r^2
@A
@ ^
2 cot r^2 A^ ^
r^2 sin
@A
er
r^2 A 1 r^2 sin^2
A +^2
r^2
@Ar @
2 cos^ r^2 sin^2
@A
e
r^2 A 1 r^2 sin^2
A + 2
r^2 sin
@Ar @
@A
e
Note that in non-cartesian coordinates,
(r^2 A)i 6 = r^2 Ai
Cylindrical Coordinates (; ; z)
Transformation
x = cos y = sin z = z
Metric coe¢ cients h = 1; h = ; hz = 1 Derivatives of the unit vectors @e @
= e ; @e @
= e
Gradient rf = @f @
e +^1
@f @
e + @f @z
ez
Divergence r A =^1
(A) +^1
@A
r A =
e e
ez
@ @
@z
A A Az Scalar Laplacian r^2 f = @
(^2) f @^2
+^1
@f @
+^1
^2
@^2 f @ ^2
(^2) f @z^2
Vector Laplacian
r^2 A r(r A) r (r A)
=
r^2 A
^2
@A
@ ^
^2 A
e
r^2 A +^2 ^2
@A
^2
A
e + r^2 Az ez
4. Special Functions
Bessel Functions Zm(x) = Jm(x); Nm(x) [or Ym(x) in some books]
J 0 (x); J 1 (x)
0 5 10 15 20
1
0
-0.25 x
y
x
y
J 0 (x) (solid line) and J 1 (x) (dashed line).
Y 0 (x); Y 1 (x)
5 10 15 20
1
0
-0.
x
y
x
y
Y 0 (x) (solid line) and Y 1 (x) (dashed line). Di§erential equation (^) d^2 d^2 +
d d +^ k
(^2) m^2 ^2
Zm(k) = 0
Wronskian Jm(x)N (^) m^0 (x) J m^0 (x)Nm(x) =
x
d dx [x
mZm(x)] = xmZm 1 (x)
Integral representations (There are many. A few are listed.)
Jm(x) =
Z 2
0
cos(m x sin )d; J 0 (x) =
Z 1
0
p^ cos(xt) 1 t^2
dt
N 0 (x) =
Z 1
1
cos(xt) p 1 t^2
dt
Nm(x) =
2 m+1x m p ( 12 m)
Z 1
1
cos xt (t^2 1)m+(1=2)^
dt
Integrals (^) Z (^1)
0
Jm(ax)dx =
a ;
Z 1
0
Nm(ax)dx =
a tan
(^) m 2
Z 1
0
J 0 (ax)J 0 (bx)dx = (^) b^2 K(a=b); K : complete elliptic integral of the Örst kind
Z (^1)
0
J 0 (ax)J 1 (bx)dx =
1 =b; b > a > 0 1 = 2 b; a = b > 0 0 ; a > b > 0
(step function)
Z (^1)
0
xJ 1 (ax)J 1 (bx)dx =
(a b) a ;^ (derivative of the above with respect to^ a) In fact for any integer m; Z (^1)
0
xJm(ax)Jm(bx)dx = (a^ ^ b) a Z (^1)
0
J+x(ax)J x(ax)dx = J+ (2a); + > 1 Z (^1)
0
x ^1 J (ax)dx =
2 ^1
2
a
(^) ; : gamma function Z (^1)
0
J (ax)J (bx) x^2 y^2 dx^ =^
i 2 J^ (by)H
(1) (ay) Z (^1)
0
e axJ 0 (bx)dx = p^1 a^2 + b^2 Z (^1)
0
e axJ (bx)dx = (
p a^2 + b^2 a) b^
p a^2 + b^2 Z (^1)
0
e axJ (bx)J (cx)dx =
p bc
Q 12
a^2 + b^2 + c^2 2 bc
; Q : Legendre function of the 2nd kind Z (^1)
0
e a^2 x^2 J 2 (px)dx =
p 2 a exp
b^2 8 a^2
I
b^2 8 a^2
Z 1
0
e a^2 x^2 x^2 J 0 (bx)dx = (^21) a 2 exp
b
2 4 a^2
Z 1
0
e a^2 x^2 J (px)J (qx)dx = (^21) a 2 exp
p
(^2) + q 2 4 a^2
I
(^) pq 2 a^2
Z 1
0
sin(ax)J (bx)dx =
pb (^2) a 2 sin sin ^1 (a=b)^ ; b > a p^ b a^2 b^2 (a+pa^2 b^2 )^ cos^
2
; a < b
Z (^1)
0
sin(ax)J 0 (bx)dx =
0 ; b > a p^1 a^2 b^2
; a > b
Sum J 0 (x) + 2
X^1
n=
J 2 n(x) = 1
X^1
n=
J n^2 (x) = 1
X^1
n=
n^2 J n^2 (x) = x
2 4
J 0 (x) + 2
X^1
n=
( 1)nJ 2 n(x) = cos x
X^1
n=
( 1)nJ 2 n+1(x) =
2 sin^ x X^1
n=
(2n + 1)J 2 n+1(x) = x 2 X^1
n=
n^2
J 2 n(2nx) = x
2 2 X^1
n=
J 2 n(2nx) =
x^2 2(1 x^2 ) X^1
n=
n^2 J 2 n(2nx) = x
(^2) (1 + x (^2) ) 2(1 x^2 )^4 X^1
n=
n^2
Z (^) x
0
J 2 n(2nt)dt = x
3 6(1 x^2 )^3
Spherical Bessel Functions zl(x) = jl(x); nl(x)
Spherical Bessel functions are elementary functions. Some low order forms are:
j 0 (x) = sin^ x x
; j 1 (x) = sin^ x^ ^ x^ cos^ x x^2
; j 2 (x) = (3^ ^ x
(^2) ) sin x 3 x cos x x^3
n 0 (x) = cosx^ x; n 1 (x) = cos^ x^ +x^2 x sin^ x; n 2 (x) = (3^ ^ x
(^2) ) cos x + 3x sin x x^3 DeÖnition jl(x)
r 2 x
Jl+ 12 (x); nl(x)
r 2 x
Nl+ 12 (x)
0 0.5 1 1.5 2 2.5 3
10
5
0 x
y
x
y
K 0 (x) (solid line) and K 1 (x) (dashed line).
DeÖnition
Im(x) = e im=^2 Jm(ix)
=
(^) x 2
m X^1
0
(x=2)^2 n n!(m + n)!
Km(x) = ( 1)m+1Im(x)
x 2
( 1)m 2
X^1
k 0
(x=2)m+2k k!(m + k)!
" (^) k X
n=
n +
kX+m
n=
n
mX 1
r=
( 1)r^
(m r 1)! r!
(^) x 2
2 r m
Di§erential equation d^2 d^2 +
d d +^ k
(^2) + m^2 ^2
Im(k) Km(k)
Wronskian I m^0 (x)Km(x) Im(x)K m^0 (x) =^1 x Series representation of Im(x)
Im(x) =
(^) x 2
m X^1
n=
m!(m + n)!
(^) x 2
2 n
For x 1 I 0 (x) ' 1 +^1 4
x^2 +
I 1 (x) ' x 2
+^1
x^3 +
Recurrence formulae
Im 1 (x) Im+1(x) =^2 m x
Im(x); Im 1 (x) Im+1(x) = 2I^0 m(x)
Km 1 (x) Km+1(x) =
2 m x Km(x);^ Km ^1 (x) +^ Km+1(x) =^ ^2 K
0 m(x)
I 00 (x) = I 1 (x); K 00 (x) = K 1 (x) Integral representation
K 0 (x) =
Z 1
0
tJ 0 (xt) t^2 + 1 dt^ =
Z 1
0
p^ cos^ xt t^2 + 1
dt
K 1 (x) = K^00 (x) =
Z 1
0
t^2 J 1 (xt) 1 + t^2 dt
K 1 = 3
2 x^3 =^2 33 =^2
p^3 x
Z 1
0
cos(t^3 + xt)dt; (Airyís integral)
Legendre Functions P lm (x); Qml (x)
Di§erential equation (1 x^2 )
d^2 dx^2 ^2 x
d dx +^ l(l^ + 1)^ ^
m^2 1 x^2
P (^) lm (x) Qml (x)
Pl(x) = (^21) ll!^ d
l dxl^ (x
(^2) 1)l (^) (Rodriguesíformula)
Ql(x) =^1 2
Pl(x) ln 1 +^ x 1 x
Wl 1 (x); x real and jxj 1
Ql(z) =^1 2
Pl(z) ln z^ + 1 z 1
Wl 1 (z) for general complex z
W 1 (x) = 0; W 0 (x) = 1; W 1 (x) =^3 2
x; W 2 (x) =^5 2
x^2 2 3
Special values (m = 0) Pl(1) = 1; Pl( 1) = ( 1)l
Pl(0) = 0 for odd l; Pl(0) = ( 1)l=^2 (l^ ^ 1)!! l!!
for even l
P (^) l^0 (1) = l(l^ + 1) 2
; P (^) l^0 (0) = (l + 1)Pl+1(0)
DeÖnition of P (^) lm (x); Qml (x) in terms of Pl(x); Ql(x) (x real, jxj 1)
P (^) lm (x) =
1 x^2
m= 2 dm dxm^ Pl(x);^ Q
m l (x) =^
1 x^2
m= 2 dm dxm^ Ql(x) For general complex argument z
P (^) lm (z) =
z^2 1
m= 2 dm dzm^
Pl(z); Qml (z) =
z^2 1
m= 2 dm dzm^
Ql(z)
Orthogonality of P (^) lm (x) Z (^1)
1
P (^) lm (x)P (^) lm 0 (x) 1 x^2 dx^ =
m
(l + m)! (l m)! mm
0
Z (^1)
1
P (^) lm (x)P (^) lm 0 (x)dx =
2 l + 1
(l + m)! (l m)! ll
0
Negative m : Yl; m(; ) = ( 1)mY (^) lm(; ) for l m 0 General form
Ylm(; ) = ( 1)
m+jmj 2
s 2 l + 1 4
(l jmj)! (l + jmj)! P^
jmj l (cos^ )e
im ; for l m l
Y 00 (; ) =
p 4
Y 10 (; ) =
r 3 4 cos^
Y 1 ; 1 (; ) =
r 3 8 sin^ e
i
Y 20 (; ) =
r 5 4
(3 cos^2 1)
Y 2 ; 1 (; ) =
r 15 8 sin^ ^ cos^ e
i
Y 2 ; 2 (; ) = 1
r 15 2
sin^2 ei^2
Orthogonality of Ylm(; ) I Ylm(; )Y (^) l (^0) m 0 (; )d = ll^0 mm^0 ; d = sin dd
Toroidal functions Pl 12 (cosh ); Ql 12 (cosh ) satisfy d^2 d^2
l^2 +^1 4
m^2 cosech^2
F () = 0
Integral representations
P (^) lm 12 (cosh ) =
( 1)m(2l 1)!! 2 m+1(2l 2 m 1)!!
Z
0
cos m (cosh + cos sinh )l+^
12 d
Qml 1 2
(cosh ) = ( 1)
m(2l 1)!! 2 m+1(2l 2 m 1)!!
Z 1
0
cosh mt (cosh + cosh t sinh )l+^12
dt
Gamma Function
DeÖnition (z) =
Z 1
0
e ttz ^1 dt
Properties
(z + 1) = z (z) ; (z) (1 z) =
sin (z) ;^ ^
z + (^12)
2 ^ z
cos (z) If z is a positive integer, z = n; (n + 1) = n!
Special values (1) = 1;
2
p ;
n + (^12)
= (2n^ ^ 1)!! 2 n
p
(x)
-5 -2.5 0 2.5 5
5
0
-2.
x
y
x
y
Gamma function (x) :
Elliptic Integrals K (k^2 ) and E (k^2 )
Complete Elliptic Integrals of the First Kind K(k^2 ) and Second Kind E
k^2
DeÖnitions
K
k^2
Z = 2
0
p^1 1 k^2 sin^2
d; E
k^2
Z = 2
0
p 1 k^2 sin^2 d; 0 k^2 1
Special values
K(0) = E (0) = 2
; lim "! 0 K (1 ") = ln
p^4 "
; E (1) = 1
K (x) =
Z = 2
0
p^1 1 x sin^2
d
E (x) =
Z = 2
0
p 1 x sin^2 d
cosh x = 1 +^1 2!
x^2 +^1 4!
x^4 +
sinh x = x +
3! x
3 +^1
5! x
tanh x = x 13 x^3 + 152 x^5 31517 x^7 +
ln(1 + x) = x 1 2
x^2 +^1 3
x^3 ; jxj < 1
InÖnite Products
Y^1
n=
1 + x
2 n^2
= sinh^ x x
Y^1
n=
1 x
2 n^2
= sin^ x x Y^1
n=
1 + x
2 (2n 1)^2
= cosh x 2 Y^1
n=
x^2 (2n 1)^2
= cos
x 2 Y^1
n=
1 + x
2 (a 2 n)^2
= cosh^ x^ ^ cos^ a 1 cos a Y^1
n=
x^2 (a 2 n)^2
cos x cos a 1 cos a
