30 I

.............................................2........................................3................................................4.......................................... 2.5.......................................... 2.6................................... 3.7.......................................... 4 2 5 2............................................ 5 2..................................... 5 2..2................................... 5 2..3....................................... 7 2..4..................................... 7 2..5..................................... 8 2..6......................... 9 2.2........................................... 0 2.2......................................... 0 2.2.2....................................... 0 2.2.3....................................... 2.2.4................................. 3 2.2.5...................................... 4 2.2.6..................................... 4 2.3........................................... 5 2.3.................................... 5 2.3.2.............................. 6 2.3.3................................... 6 2.3.4...................................... 7 3 8 3. Γ B.......................................... 8 3.. Γ........................................... 8 3..2 B........................................... 9 3.2....................................... 9 3.2...................................... 9 3.2.2........................................... 20 3.2.3........................................... 20 3.2.4 (Rodrigues)............................. 22 3.2.5 (Goursat).............................. 22 3.3...................................... 23 3.3. (Jacobi).................................. 23 3.3.2 (Gegenbauer).......................... 23 3.3.3 (Legendre)............................. 24 i

3.3.4 (Chebyshev)............................ 27 3.4.................................... 28 3.4. (Sonine)................................. 28 3.4.2 (Laguerre)............................. 29 3.4.3 (Laguerre).............................. 30 3.4.4................................... 3 3.5 (Hermite)..................... 3 4 33 4........................................... 33 4.2 (Sturm-Liouville).................... 34 4.3........................................... 34 4.4................................ 35 4.5....................................... 37 4.5.................................... 37 4.5.2................................... 38 4.5.3............................ 39 4.5.4............................ 40 4.5.5................................ 40 4.5.6............................. 4 4.6..................................... 4 4.6................................... 4 4.6.2....................................... 42 A 45 ii

. N 2 a b a 2 a =., b = b 2., a N b N a b N a b = a i b i = a b + a 2b 2 + + a Nb N (.) i= a b = 0 a b a () a a = a a (.2).2 N {a, a 2,..., a N } α a + α 2 a 2 + + α N a N = 0 (.3) {α, α 2,..., α N } α = α 2 = = α N = 0 N (.3) {α, α 2,..., α N } α = α 2 = = α N = 0 N. ( ) ( ) 0 () 2 a = a 2 = 0 ( ) ( ) 2 (2) 2 a = a 2 = 0 0 ( ) ( ) ( ) 0 (3) 3 a = a 2 = a 3 = 0.3 N N {a, a 2,..., a N } {a i }

{a, a 2,..., a N } x N x = α a + α 2 a 2 + + α N a N = α i a i (.4) i= α i {a i } {a i }.4 ( ) {a i } x (.4) x (α, α 2,..., α N ) 2 ( ) ( ) 0 e = e 2 = (x, y) x 0 x = xe + ye 2 = (x, y) ( ) x.2 (3, 2) ( ) ( ) () 2 a = a 2 = ( ) ( ) (2) 2 b = 0 b 2 = () a a 2 = 0 a a 2 a = a 2 = 2 (2) b b 2 y.5 N N {a i } a i a j = 0 (for i j) (.5) a = a 2 = = a N = (.6) a i a j = δ ij (.7) 2

N N x (.4) {a i } {α i } α i = a i x (.8) {a i } x N x = (a i x)a i (.9).3 (.8) i=.6 () () N {a i } {e i }. e = a / a e 2. a 2 = a 2 (a 2 e )e e 2 = a 2/ a 2 e 2 3. n n + N {e, e 2,..., e n, a n+,..., a N } n + a n+ = a n+ (a n+ e )e (a n+ e 2 )e 2 (a n+ e n )e n n = a n+ (a n+ e i )e i (.0) i= e n+ = a n+/ a n+ (.) 4. 3 n + = N a.4 3 e n+ e j (j n) ( ) ( ).5 2 2 a = 0 a 2 = 2 {e, e 2 } 0.6 3 3 a = a 2 = 0 a 3 = 0 () {e, e 2, e 3 } (2) (e i e j = δ ij ) 3

.7 (N ) 4

2 2. (x n ) 2.. f(x) () f(x) ( ) f(x) f (x) f(x) 2..2 π, π 2π (f(x) = f(x + 2π)) f(x) {cos nx} {sin nx} f(x) a 0 2 + (a n cos nx + b n sin nx) (2.) n= a n (n =, 2, 3,... ) b n (n =, 2, 3,... ) = f(x) f(x) (f(x + 0) + f(x 0)) (2.2) 2 (2.) a n b n a n = π b n = π π π π π f(x) cos nx (n 0) (2.3) f(x) sin nx (n ) (2.4) 5

π π π π π π 2. (2.5) (2.7) 2.2 (2.5) (2.7) (2.3) (2.4) 2.3 0 π < x 0 () f(x) = 0 < x π x π < x 0 (2) f(x) = x 0 < x π π < x 0 (3) f(x) = 0 < x π ( ) () f(x) 2 + (3) f(x) n= n= cos mx cos nx = πδ mn (n, m ) (2.5) sin mx sin nx = πδ mn (n, m ) (2.6) cos mx sin nx = 0 (n, m ) (2.7) nπ ( ( )n ) sin nx (2) f(x) = π 2 + 2 n 2 π (( )n ) cos nx 2 nπ ( ( )n ) sin nx 2.3 (2) sin nx 0 (3) cos nx 0 cos nx sin nx n= 2.3() 3 (a) 0 (b) 00 (c) ( ) 0.8 0.6 0.4 0.2 3 2 2 3 0.8 0.6 0.4 0.2 3 2 2 3 0.8 0.6 0.4 0.2 3 2 2 3 (a) (b) (c) (d) (c) 2π 6

0.8 0.6 0.4 0.2 0 5 5 0.3.2. 0.9 0.8 0.7 0.5.5 2 2.5 3 (d) (e) 2..3 (e) 2.3() 000 x = π 8% ( ) ( ) 2..4 2π L L/2, L/2 L (f(x) = f(x + L)) f(x) (2.) (2.3) (2.4) x 2π L x f(x) a 0 2 + (a n cos 2πnx L n= + b n sin 2πnx L ) (2.8) a n = 2 L b n = 2 L L/2 L/2 L/2 L/2 f(x) cos 2πnx (n 0) (2.9) L f(x) sin 2πnx (n ) (2.0) L L = 2π (2.) (2.3) (2.4) 2.4 () f(x) = x, < x + x < x 0 (2) f(x) = x 0 < x ( ) () f(x) n= (f(x + 0) + f(x 0)) (2.) 2 ( ) n 2 nπ sin nπx (2) f(x) = 2 + 2 n 2 π 2 ( ( )n ) cos nπx n= 7

2..5 π, π 2π f(x) {sin nx, cos nx} {e inx } e inx = cos nx + i sin nx (2.2) e inx = cos nx i sin nx (2.3) cos nx = 2 (einx + e inx ) (2.4) sin nx = 2i (einx e inx ) (2.5) (2.) f(x) a 0 2 + (a n ib n )e inx + (a n + ib n )e inx ) (2.6) n= a 0 2 = c 0, a n ib n 2 = c n, c n c n (c n = c n ) (2.6) a n + ib n 2 = c n (2.7) f(x) n= c n e inx (2.8) c n = π f(x)e inx (2.9) 2π π n ({sin nx, cos nx} 0 ) (2.9) π π e inx e imx = 2πδ nm (2.20) 2.5 (2.20) 2.6 (2.20) (2.9) 2.7 2.3 () (2) 2.3 ( ) () f(x) 2 + (3) f(x) n= n 0 n= n 0 i nπ (( )n )e inx i 2nπ (( )n )e inx (2) f(x) = π 2 + n= n 0 n 2 π (( )n )e inx 8

L/2, L/2 L f(x) f(x) c n = L L/2 n= L/2 c n e i 2πnx L (2.2) 2πnx i f(x)e L (2.22) 2..6 f(x) g(x) (f(x) g(x)) (f(x) g(x)) = 2π π π f(x) g(x) (2.23) f(x) g(x) (f(x) g(x)) 0 f(x) g(x) f(x) (f(x) f(x)) f(x) (2.23) 2π f(x) e inx ( ) x a n a n = e inx x = f(x) (2.23) (a m a n ) = π e imx e inx = δ mn (2.24) 2π π (.7) {e inx } (2.2) (2.22) x = f(x) = c n e inx π = f(x)e inx e inx 2π n= n= π π = (e inx ) f(x) e inx = (a n x)a n (2.25) 2π π n= n= (.9) {e inx } {e inx } ( ) (2.8) ( 2) (2.23) 2π (f(x) g(x)) = { 2π e inx } π π f(x) g(x) (2.26) ( 3) {x n } {x n } {x n } 9

2.2 X 2.2. f(x) f(x)δ(x x 0 ) = f(x 0 ) (2.27) δ n (x) (δ(x) = lim n δ n (x)) δ(x x 0 ) = (2.28) δ(x x 0 ) x = x 0 (δ nm ) {. δ(x) = lim δ n x /2n n(x), δ n (x) = n 0 x > /2n 2. δ(x) = lim δ sin nx n(x) = lim n n πx 3. δ(x) = lim n δ n(x) = lim n n 2 π e nx 2.8 (2.28) 2.9 () δ(x) = δ( x) (2) δ(ax) = a δ(x) (3) δ(x 2 a 2 ) = 2a {δ(x a) + δ(x + a)} (a > 0) 2.2.2 L/2, L/2, L ( ) 2π L = k (2.29) 0

(2.2) (2.22) f(x) = c n = L n= L/2 L/2 c n e in kx (2.30) e in kx (2.3) L k n k n k = k (2.30) n k (L ) n k k dn dk k (2.30) dk = L dk (2.32) k 2π (2.30) n= f(x) = L 2π (2.3) c n c n e ikx dk (2.33) f(x) = f(t)e ikt dt e ikx dk (2.34) 2π k f(t) k f(x) = f(t) e ik(x t) dk dt (2.35) 2π (2.34) (2.35) (2.27) (2.35) 2π eik(x t) dk δ(t x) δ(x) = δ( x) δ(x x 0 ) = e ik(x x0) dk (2.36) 2π 2.2.3 (2.34) 2 F (k) = f(x) = 2π f(x)e ikx = F{f(x)} (2.37) F (k)e ikx dk = F {F (k)} (2.38)

() ( ) F (k) f(k) ( ) /2π F (k) = f(x) = 2π 2π f(x)e ikx = F{f(x)} (2.39) F (k)e ikx dk = F {F (k)} (2.40) k 2πk (2.37) (2.38) (2.37) (2.38) (2.39) (2.40) (2.34) f(x) = F {F{f(x)}} (2.4) f(x) 2.0 { x a f(x) = 0 x > a 2 sin ka ( ) k 2. () δ(x) (2) () (2.36) 2.2 f(x) = a/πe ax2 ( ) F (k) = I n = I n = I n = a π e ax2 e ikx = = a π x 2n e ax2 x 2n ( (2n )!! π (2a) n a e ax2 2a a π e ax2 cos kx = ) ) (e = x 2n ax2 e ax2 (cos kx + i sin kx) 2a = 2n 2a I n = 2n 2a a π 2n 3 2a e ax2 n=0 + 2n 2a a k 2n π 2n! x2n e ax 2 a k2n (2n )!! π = ( )n π 2n! (2a) n a = ( ) n (2n ) (2n 3) 3 k 2n 2n (2n )(2n 2) (2n 3) 3 2 (2a) n = ( ) n (kx)2n ( ) 2n! x 2n e ax2 3 2a 2a I 0, I 0 = e ax2 = π a ( ) n k 2n 2n(2n 2) 4 2 (2a) n = ( )n k 2n n!(4a) n 2

( ) = n=0 ( ) n k 2n n!(4a) n F (k) = e k2 /4a = n=0 n! ) ( k2 = e k2 /4a 4a 2. 2.2 2.2 II { 0 x < 0 2.3 θ(x) = θ(x) 0 x { 0 x < 0 () θ ϵ (x) e ϵx 0 x θ ϵ(x) θ(x) = lim ϵ 0 θ ϵ (x) θ(x) θ(k) θ(k) = lim ϵ 0 ϵ + ik (2) () θ(x) = lim 2π (3) (2) ( dθ(x) ϵ 0 e ikx dk ϵ + ik = δ(x)) 2.2.4 f(x) (, ) f(x) f (x) x 0 f (x)e ikx = ik f(x)e ikx (2.42) F{f (x)} = ikf{f(x)} (2.43) ik n dn f(x) n = f (n) (x) x 0 f (n) (x)e ikx = (ik) n f(x)e ikx (2.44) F{f (n) (x)} = (ik) n F{f(x)} (2.45) n (ik) n 2.4 (2.42) (2.43) 2.5 (2.44) (2.45) 3

2.2.5 (, ) 2 f(x) g(x) f g = f(x u)g(u)du (2.46) f g f(x) g(x) f g = f(x u)g(u)du = F (k)g(k)e ikx dk (2.47) 2π f g = F {F (k)g(k)} = F {F{f(x)}F{g(x)}} (2.48) F{f g} = F (k)g(k) = F{f(x)}F{g(x)} (2.49) ( ) (2.37) (2.38) (2.39) (2.40) (2.47) f(x u)g(u)du = F (k)g(k)e ikx dk (2.50) (2.48) (2.49) f g = 2π 2.6 (2.47) f(x u)g(u)du (2.5) 2.2.6 f(x)g (x) = F (k)g (k)dk (2.52) 2π g (x) G (k) g(x) G(k) f(x) = g(x) f(x) 2 = F (k) 2 dk (2.53) 2π g(x) = f(x)e ik x 4

f(x) 2 e ik x = F (k)f (k k )dk (2.54) 2π 2.7 (2.52) 2.8 (2.54) 2.9 2.0 sin 2 x x 2 2.3 2.3. f(x) F (s) = 0 f(x)e sx = L{f(x)} (2.55) F (s) f(x) s F (s) (2.55) f(x) f(x) F (s) f(x) = L {F (s)} (2.56) f(x) = L {L{f(x)}} (2.57) f(x) = L {F (s)} = 2πi γ+i γ i F (s)e sx ds (2.58) II (2.55) ( ) x M e s0x f(x) M (2.59) s 0 s > s 0 x e sx 5

2.3.2 f(x) F (s) δ(x) + 0 (2.60) s s > 0 (2.6) x n n! s n+ s > 0 (2.62) e ax s a s > a (2.63) xe ax (s a) 2 s > a (2.64) s cosh ax s 2 a 2 s > a (2.65) a sinh ax s 2 a 2 s > a (2.66) s cos ax s 2 + a 2 s > 0 (2.67) a sin ax s 2 + a 2 s > 0 (2.68) e bx cos ax s b (s b) 2 + a 2 s > b (2.69) e bx sin ax a (s b) 2 + a 2 s > b (2.70) x cos ax x sin ax s 2 b 2 (s 2 + a 2 ) 2 s > 0 (2.7) 2as (s 2 + a 2 ) 2 s > 0 (2.72) 2.20 (2.60) (2.6) (2.63) (2.65)-(2.68) 2.3.3 L{af(x) + bg(x)} = al{f(x)} + bl{g(x)} (2.73) L{f(λx)} = F (s/λ) λ (2.74) L{e ax f(x)} = F (s + a) (2.75) L{f(x a)} = e as F (s) (2.76) I L{f (x)} = s{f(x)} f(0) = sf (s) f(o) (s > 0) (2.77) II L{f (x)} = s 2 {f(x)} sf(0) f (0) = s 2 F (s) sf(0) f (0) (s > 0) (2.78) d n F (s) ds n = F (n) (s) = L{( x) n f(x)} (2.79) { } f(x) F (t)dt = L (2.80) x s 6

2.2 (2.73) (2.80) 2.22 (2.75) (2.69) (2.70) 2.23 (2.79) (2.62) (2.64) (2.7) (2.72) 2.3.4 f(x) g(x) f g f g = 0 f(x u)g(u)du (2.8) { } L{f g} = L f(x u)g(u)du = F (s)g(s) = L{f(x)}L{g(x)} (2.82) 0 2 2.24 (2.82) 7

3 3. Γ B 3.. Γ x > 0 Γ(x) = Γ() 0 e t t x dt (3.) Γ Γ(x + ) = xγ(x) (3.2) x n Γ(n + ) = n! (3.3) Γ (n!) Γ() = Γ(/2) = π 3. (3.2) 3.2 Γ() = Γ(/2) = π x < 0 (3.2) x + n > 0 n Γ(x) = Γ(x + n) x(x + )... (x + n ) (3.4) x (Stirling) x Γ(x + ) 2πxx x e x (3.5) log Γ(x + ) x log x x + log 2πx x log x x (3.6) 20 Γ(x + ) ( 3, 4) 2πxx x e x (0, 4) Γ(x + ) 2πxx x e x 0-3 -2-2 3 4-0 -20 8

3..2 B x > 0 y > 0 B(x, y) = B() 0 t x ( t) y dt (3.7) B Γ B(x, y) = Γ(x)Γ(y) Γ(x + y) (3.8) 3.3 (3.8) 3.2 3.2. f(x) x f(x) = a 0 + a x + + a n x n + a n x n (3.9) f(x) x n a, b w(x) (f(x), g(x)) = b a w(x)f (x)g(x) (3.0) 0 f(x) g(x) n ϕ n (x) n ϕ n (x) = a n kx k = a n 0 + a n x + + an x n n + a n nx n (3.) k=0 a n k n k ( ) (a k n ) {ϕ n (x)} = {ϕ 0 (x), ϕ (x), ϕ 2 (x),... } (ϕ m (x), ϕ n (x)) = b a w(x)ϕ m(x)ϕ n (x) = δ mn w n (3.2) {ϕ n (x)} ϕ m(x) ϕ m (x) w n = ( ) w n ( ϕ n (x)/ w n ϕ n (x) ) 9

( ) (3.2) ϕ m(x) = ϕ m (x) {, x, x 2, x 3,... } 3.4, w(x) = {, x, x 2, x 3 } 3 n x m (m n ) b a w(x)ϕ n (x)x m = 0 (0 m n ) (3.3) 3.5 (3.3) 3.2.2 w(x) a, b n b a w(x)xn (3.) (3.2) a n n 3.2.3 ϕ n (x) = (α n x β n )ϕ n (x) γ n ϕ n 2 (x) (3.4) 3 α n = a n n a n n (xϕ n (x), ϕ n (x)) β n = α n (ϕ n (x), ϕ n (x)) α n (ϕ n (x), ϕ n (x)) γ n = α n (ϕ n 2 (x), ϕ n 2 (x)) (3.5) ( ) ϕ n (x) ϕ n (x) ϕ n (x) x an n ϕ n (x) = a n 0 + a n x + + a n n x n + a n nx n ϕ n (x) = a n 0 + a n x + + a n n xn a n n ϕ n (x) n n ϕ k (x) ϕ n (x) x an n a n n n ϕ n (x) = b k ϕ k (x) (3.6) k=0 20

ϕ j (x) (j n 3) ( ϕ n (x) x an n a n n a n n a n n ϕ n (x), ϕ j (x) (xϕ n (x), ϕ j (x)) = ) an n a n n = b j (ϕ j (x), ϕ j (x)) (3.7) (ϕ n (x), xϕ j (x)) (3.8) xϕ j (x) n 2 0 (3.7) b j = 0 (j n 3) (3.9) (3.6) k = n n 2 ϕ n (x) x an n a n n ϕ n (x) = b n ϕ n (x) + b n 2 ϕ n 2 (x) (3.20) α n = a n n/a n n, β n = b n, γ n = b n 2 (3.2) ϕ n (x) = (α n x β n )ϕ n (x) γ n ϕ n 2 (x) (3.22) b n b n 2 β n γ n (3.22) ϕ n (x) 0 = α n (xϕ n (x), ϕ n (x)) β n (ϕ n (x), ϕ n (x)) β n = α n (xϕ n (x), ϕ n (x)) (ϕ n (x), ϕ n (x)) (3.22) ϕ n 2 (x) (3.23) 0 = α n (xϕ n (x), ϕ n 2 (x)) γ n (ϕ n 2 (x), ϕ n 2 (x)) γ n = α n (xϕ n (x), ϕ n 2 (x)) (ϕ n 2 (x), ϕ n 2 (x)) (3.24) (xϕ n (x), ϕ n 2 (x)) = (ϕ n (x), xϕ n 2 (x)) = (ϕ n (x), a n 2 n 2 xn +... ) = an 2 a n n ϕ n (x) ϕ n (x) = a n n xn + n 2 (ϕ n (x), a n n xn +... ) = an 2 k=0 an k x k n 2 a n n xn = ϕ n (x) a n 2 k x k (3.25) ( ) (xϕ n (x), ϕ n 2 (x)) = an 2 n 2 a n ϕ n (x), ϕ n (x) a n k x k n (3.24) γ n γ n = k=0 k=0 α n (ϕ n (x), ϕ n (x)) α n (ϕ n 2 (x), ϕ n 2 (x)) a n n = an 2 a n n (ϕ n (x), a n n xn ) (3.25) (ϕ n (x), ϕ n (x)) (3.26) 2

3.6 (3.4) (3.5) w(x) =, a n n 3.2.4 (Rodrigues) G (k) n ϕ n (x) = (a) = G (k) n c n d n G n (x) w(x) n (3.27) (b) = 0 G (k) n (x) G n (x) k ( ϕ n (x) ) ϕ n (x) n (3.27) 3.7 (3.27) ϕ n (x) n d n+ c n d n G n (x) n+ w(x) n = 0 (3.28) G n (x) w(x) {ϕ n (x)} 3. 2. 3. 3.2.5 (Goursat) II (Goursat) f(z) n f (n) (z) f (n) (z) = n! f(t) dt (3.29) 2πi C (t z) n+ C t = z (3.27) ϕ n (z) = n! G n (t) dt (3.30) 2πi C w(z)(t z) n+ 22

3.3 a, b, a, b, (Jacobi) (Legendre) (Chebyshev) ( ) 0, 3.3. (Jacobi) {P n (α,β) (x)} w(x) = ( x) α ( + x) β, P n (α,β) (x) = ( )n 2 n n! w(x) Dn w(x)( x 2 ) n, = ( )n 2 n n! ( x) α ( + x) β D n ( x) n+α ( + x) n+β, (3.3) D = d w n = w(x)p n (α,β) (x)p m (α,β) (x) = w n δ nm, (3.32) 2 α+β+ 2n + α + β + Γ(n + α + )Γ(n + β + ). (3.33) n!γ(n + α + β + ) 2(n + )(n + α + β + )(2n + α + β)p (α,β) n+ (x) = (2n + α + β + ){(2n + α + β)(2n + α + β + 2)x + (α 2 β 2 )}P n (α,β) (x) 2(n + α)(n + β)(2n + α + β + 2)P (α,β) n (x) (3.34) P (α,β) 0 (x) =, P (α,β) (x) = {(α + β + 2)x + (α β)} (3.35) 2 3.3.2 (Gegenbauer) {Cn(x)} λ α = β = λ /2 w(x) = ( x 2 ) λ 2 Cn(x) λ = Γ(λ + 2 )Γ(n + 2λ) (λ Γ(2λ)Γ(n + λ + 2,λ 2 ) 2 )P n (x) (3.36) 23

Cn(x) λ = ( )n Γ(λ + 2 )Γ(n + 2λ) 2 n n!γ(2λ)γ(n + λ + 2 ) w(x) Dn w(x)( x 2 ) n (3.37) w(x)c λ n (x)c λ m(x) = π2 2λ Γ(n + 2λ) n!(λ + n)γ 2 (λ) δ nm (3.38) (n + )C λ n+(x) = 2(λ + n)xc λ n(x) (2λ + n )C λ n (x) (3.39) C λ 0 (x) =, C λ (x) = 2λx (3.40) 3.3.3 (Legendre) P n (x) w(x) =, α = β = 0 ν = /2 P n (x) = P (0,0) n (x) = C /2 n (x), (3.4) (3.42) P n (x) = ( )n 2 n n! Dn ( x 2 ) n, (3.42) P n (x) = 2 n n 2 m=0 x x n/2 n n/2 n (n )/2 (3.43) P n (x) n n 4 n ( ) m (2n 2m)! m!(n m)!(n 2m)! xn 2m (3.43) 0.5 0.5 n n n n + n + 0.5 0.5 P 0 x P x P 2 x P 3 x P 4 x 24

3.8 (3.43) P m(x)p n (x) = 2 2n + δ nm, (3.44) 3.9 (3.44) P 0 (x) = P (x) = x (n + )P n+ (x) = (2n + )xp n (x) np n (x), (3.45) 3.0 (3.45) / 2xt + t 2 t t = 0 = P n (x)t n (3.46) 2xt + t 2 n=0 / 2xt + t 2 3. (3.46) ( ) f(x) = x x = f(x) n f (n) (x) = (2n)! 2 2n n! ( x) 2 n f (0) (x) = ( x) 2 f () (x) = 2 ( x) 3 2 f (k+) (x) = d (2k)! 2 2k k! ( x) 2 k = (2k)! 2 2k k! = (2k)! 2 2k k! = 2k + ( x) 2 (k+) = (2k)! 2 2 2k k! (2(k + ))! 2 2(k+) (k + )! ( x) 2 (k+) n = 0, n = k ( 2 k ) ( x) 2 (k+) (2k + 2)(2k + ) ( x) 2 (k+) 2(2k + 2) n = k + f(x) (2n)! f(x) = = x 2 2n (n!) 2 xn x 2xt t 2 n=0 = (2n)! 2xt + t 2 2 2n (n!) 2 (2xt t2 ) n = = = n=0 (2n)! 2 2n (n!) 2 tn n n=0 m=0 n=0 n nc m (2x) n m ( t) m = m=0 n=0 n=0 ( ) m (2n)! 2 2n n!m!(n m)! (2x)n m t n+m ( ) (2n)! 2 2n (n!) 2 tn (2x t) n (2n)! 2 2n (n!) 2 tn n ( ) m n! m!(n m)!) (2x)n m ( t) m m=0 25

n + m = n ( ) = = = n 2 n =0 m=0 n 2 n =0 m=0 P n (x)t n n =0 ( ) m (2n 2m)! 2 2(n m) (n m)!m!(n 2m)! (2x)n 2m t n ( ) m (2n 2m)! 2 n (n m)!m!(n 2m)! xn 2m t n 3.2 (3.46) (3.42) (3.29) 3.3 (3.46) (3.45) z = a q (r, θ, φ) ϕ ϕ r ϕ = 4πϵ 0 q r (3.47) r r = r 2 2ar cos θ + a 2 ϕ = q 4πϵ 0 r 2 a r cos θ + ( a r ) 2 (3.48) (3.46) ϕ ϕ = q ( a ) n P n (cos θ) (3.49) 4πϵ 0 r r n=0 z = a q z = a q (r, θ, φ) ϕ ϕ ϕ = q ( ) 4πϵ 0 r r 2 { q ( a ) n ϕ = P n (cos θ) 4πϵ 0 r r = 2q 4πϵ 0 r n=0 P (cos θ) a r + P 3(cos θ) n=0 (3.50) } ( P n (cos θ) a ) n = 2q r 4πϵ 0 r ( a r ) 3 +... n=0 ( a ) 2n+ P 2n+ (cos θ) r (3.5) 26

3.3.4 (Chebyshev) {T n (x)}, w(x) = / x 2 {P n (α,β) (x)} α = β = /2 {C λ n(x)} λ 0 T n (x) = 22n (n!) 2 (2n)! P n ( /2, /2) (x) = n 2 lim λ 0 Γ(λ)Cλ n(x) (3.52) T n (x) = ( )n (2n )!! w(x) Dn w(x)( x 2 ) n, w(x) = x 2 (3.53) (3.53) T n (x) = n 2 n 2 m=0 ( ) m (n m )! (2x) n 2m (3.54) m!(n 2m)! x x n/2 n n/2 n (n )/2 (3.54) T n (x) n n 4 n n n n n + n + 3.4 (3.54) 0.5 0.5 0.5 0.5 T 0 x T x T 2 x T 3 x T 4 x w(x)tm(x)t π n = 0 n (x) = w n δ nm,, w n = π/2 n (3.55) 3.5 (3.55) T 0 (x) = T (x) = x T n+ (x) = 2xT n (x) T n (x), (3.56) 3.6 (3.56) 27

t2 2xt+t t t = 0 2 t 2 2xt + t 2 = T 0(x) + 2 T n (x)t n (3.57) n= t2 2xt+t 2 3.7 (3.57) 3.8 (3.57) (3.53) (3.29) T n (cos θ) = cos nθ (3.58) ((cos θ + i sin θ) n = cos nθ + i sin nθ) T n (x) = 2 (x + i x 2 ) n + (x i x 2 ) n (3.59) 3.9 (3.58) sin cos 3.4 3.4. (Sonine) 0, a, b ϕ n (x) = K n w(x) Dn w(x)(x a) n (x b) n, w(x) = (x a) α (x b) β (3.60) 0, a 0, b x 2x/β β S α n (x) = lim β P (α,β) n ( 2x β ) (3.6) {S α n (x)} 28

Sn α (x) = n! w(x) Dn w(x)x n, w(x) = e x x α (3.62) 3.20 (3.6) (3.62) (3.62) S α n (x) = n k=0 ( ) k (n + α)! (n k)!(α + k)!k! xk (3.63) 0 w(x)s α m (x)s α n (x) = (n + α)! δ nm (3.64) n! 3.2 (3.64) (n+)s α n+(x) = (2n+α+ x)s α n (x) (n+α)s α n (x), S α 0 (x) =, S α (x) = α + x (3.65) 3.22 (3.65) xz exp z ( z) α+ = Sn α (x)z n (3.66) n=0 3.23 (3.66) ( ) Methematica LaguerreL 3.4.2 (Laguerre) L m n (x) = ( ) m n!s m n m(x) (3.67) n m 29

L m n (x) = ( ) m n! (n m)! w(x) Dn m w(x)x n m, w(x) = e x x m (3.68) (3.68) n m L m n (x) = ( ) m n! ( ) k n! (n m k)!(m + k)!k! xk (3.69) k=0 0 w(x)l m n (x)l m l (x) = (n!)3 (n m)! δ nl (3.70) 3.24 (3.64) (3.70) (n + m)l m n+(x) = (n + )(2n m + x)l m n (x) n 2 L n (x), L m m(x) = ( ) m m!, L m m+(x) = ( ) m (m + )!(m + x) (3.7) 3.25 (3.65) (3.7) exp xz ( ) m z ( z) m+ = (n + m)! Lm n+m(x)z n (3.72) n=0 3.26 (3.66) (3.72) 3.4.3 (Laguerre) m = 0 {L n (x)} L n (x) = L 0 n(x) (3.73) L n (x) = w(x) Dn w(x)x n, w(x) = e x (3.74) L n (x) = n! n k=0 ( ) k n! (n k)!k!k! xk (3.75) 30

(3.69) L m n (x) = ( ) m d L n (x) (3.76) m = 0 0 w(x)l n(x)l l (x) = (n!) 2 δ nl (3.77) L n+ (x) = (2n + x)l n (x) n 2 L n (x), L 0 (x) =, L (x) = x (3.78) (e /2x L n (x)/n!) n n 5 0 5 20 0.8 0.6 0.4 0.2 0.2 0.4 L 0 x L x L 2 x L 3 x L 4 x 3.4.4 n l R l n(r) a B Rn(r) l e ρ/2 ρ l L 2l+ 2r n+l (ρ), ρ = (3.79) na B 3.5 (Hermite),, (3.60) a, b α = β, x x/ α α H n (x) = lim α 2n n!α n/2 P n (α,α) ( x ) (3.80) α {H n (x)} H n (x) = ( ) n w(x) Dn w(x), w(x) = e x 2 (3.8) 3

( ) w(x) = e x2 /2 3.27 (3.80) (3.8) (3.8) n 2 H n (x) = ( ) m n! m!(n 2m)! (2x)n 2m (3.82) m=0 (e /2x2 H n (x)/ 2 n n!) 3.28 (3.82) 6 4 2 2 4 6 H 0 x H x H 2 x H 3 x H 4 x w(x)h m(x)h n (x) = 2 n n! πδ mn (3.83) 3.29 (3.83) H n+ (x) = 2xH n (x) 2nH n (x), H 0 (x) =, H = 2x (3.84) 3.30 (3.84) e 2xt t2 = n=0 n! H n(x)t n (3.85) 3.3 (3.85) 32

4 4. 2 2 p 0 (x) d2 2 u(x) + p (x) d u(x) + p 2(x)u(x) = p 0 (x) d2 2 + p (x) d + p 2(x) u(x) = Lu(x) = 0 (4.) L L L L = d2 2 p 0(x) d p (x) + p 2 (x) (4.2) L = p 0 (x) d2 2 + (2p 0(x) p (x)) d + (p 0(x) p (x) + p 2 (x)) (4.3) L = L 2p 0(x) p (x) = p (x) p 0(x) p (x)+p 2 (x) = p 2 (x) p 0(x) = p (x) (4.4) L = L p 0 (x) = p(x) p 2 (x) = q(x) (4.) Lu(x) = Lu(x) = d p(x) du(x) + q(x)u(x) (4.5) p 0(x) exp x p(t) p 0(t) dt ( ) x p 0 (x) exp p (t) p 0 (t) dt Lu(x) x p (t) = exp p 0 (t) dt u (x) + p (x) p 0 (x) exp x p (t) p 0 (t) dt u (x) + p x 2(x) p 0 (x) exp p (t) p 0 (t) dt u(x) { } d x exp p(t) p dt 0(t) u x (x) = exp p(t) p dt 0(t) u (x) + p(x) p exp x p(t) 0(x) p dt 0(t) u (x) x p 0 (x) exp p (t) p 0 (t) dt Lu(x) = d { x p (t) exp p 0 (t) dt du(x) } + p x 2(x) p 0 (x) exp p (t) p 0 (t) dt u(x) 33

4.2 (Sturm-Liouville) a, b p(x) q(x) w(x) x p(x) > 0 w(x) > 0 d p(x) du(x) + q(x)u(x) + λ n w(x)u(x) = 0 (4.6) λ n 2 2 A A 2 B B 2 A u(a) + A 2 u (a) = 0 B u(b) + B 2 u (b) = 0 λ n λ n 2 4. 4. 4. (4.7) 4.2 ( ) a, b w(x) 2 u (x) u 2 (x) b a w(x)u (x)u 2 (x) = 0 (4.8) 4.2 4.2 4.3 L 2 u(x) v(x) b v (x)lu(x) = b a a u(x)lv (x) (4.9) 4.3 (4.9) b a f (x)g(x) (f, g) (4.9) (v, Lu) = ( Lv, u) = (Lv, u) A 2 v u (v, Au) = ( t A v, u) = (Av, u) 34

A L A = t A L = L 4. 4.2 4.4 ϕ n (x) = K n w(x) Dn w(x)x(x) n (4.0) X(x) x 2 (4.0) X(x) d2 u 2 + K ϕ (x) du + λ nu = 0 λ n = n(k a + n X (x)) 2 (4.) a ϕ (x) x w(x) K ϕ (x)w(x) = Dw(x)X(x) d w(x)x(x) du + λ n w(x)u = 0 (4.2) w(x) 4.4 (4.) ( ) n+ D n+ XD wx n = n+c k D k X D n+ k D wx n k=0 k 0 2 ( X 2 ) = XD n+2 wx n + (n + )X D n+ wx n (n + )n + X D n wx n 2 = XD 2 D n wx n + (n + )X D D n wx n (n + )n + X D n wx n 2 { D n wx n = K n wϕ n } = K n XD 2 wϕ n + (n + )X (n + )n D wϕ n + X wϕ n () 2 D n+ XDwX n = D n+ XDwX X n = D n+ X n DwX + (n )X n wx DwX = K wϕ = D n+ X n K wϕ + (n )X wx n = D n+ {K ϕ + (n )X }wx n n+ = n+c k D k K ϕ + (n )X D n+ k wx n k=0 k = 0, = {K ϕ + (n )X }D n+ wx n + (n + ){K ϕ + (n )X }D n wx n 35

D n wx n = K n wϕ n = K n {K ϕ + (n )X }Dwϕ n + (n + ){K ϕ + (n )X }wϕ n (2) ()=(2) ( XD 2 wϕ n + (2X K ϕ )Dwϕ n (n + ) K ϕ + n 2 ) X wϕ n = 0 2 Dwϕ n = wϕ n + w ϕ n D 2 wϕ n = wϕ n + 2w ϕ n + w ϕ n wxϕ n + (2w X + 2wX K wϕ )ϕ n ( + w X + 2w X K w ϕ (n + ) K ϕ + n 2 ) X w ϕ n = 0 2 K wϕ = wx + w X K (w ϕ + wϕ ) = w { X + 2w X + wx X d2 2 ϕ d n + K ϕ ϕ n + λ n ϕ n = 0, λ n = n K ϕ + n } { X = n K a + n } X 2 2 λ n {Pn α,β (x)} d ( x 2 ) d2 u du + {β α (α + β + 2)x} + n(n + α + β + )u = 0 (4.3) 2 ( x) α+ ( + x) β+ du + n(n + α + β + )( x) α ( + x) β u = 0 (4.4) {P n (x)} ( x 2 ) d2 u 2 2xdu + n(n + )u = 0 (4.5) d ( x 2 ) du + n(n + )u = 0 (4.6) {T n (x)} ( x 2 ) d2 u 2 xdu + n2 u = 0 (4.7) d x 2 du + n 2 x 2 u = 0 (4.8) 36

{S α n (x)} x d2 u + (α + x)du + nu = 0 (4.9) 2 d e x x α+ du + ne x x α u = 0 (4.20) {L m n (x)} x d2 u + (m + x)du + (n m)u = 0 (4.2) 2 d e x x m+ du + (n m)e x x m u = 0 (4.22) {H n (x)} d 2 u 2 2xdu + 2nu = 0 (4.23) d e x2 du + 2ne x2 u = 0 (4.24) 4.5 4.5. 2 Lu = p 0 (x) d2 u 2 + p (x) du + p 2(x)u = 0 (4.25) u(n, x) Lu = 0 Lu(n, x) = f(n, x) d + g(n, x) f(n, x) d + g(n, x) u(n, x) + c(n)u(n, x) (4.26) = f(n ± l, x) d + g(n ± l, x) f(n ± l, x) d + g(n ± l, x) u(n, x) + c(n ± l)u(n, x) (4.27) 2 T = f(n, x) d + g(n, x) T = f(n ± l, x) d (4.28) + g(n ± l, x) 37

T u(n, x) = λ n u(n l, x) T u(n, x) = λ n u(n ± l, x) (4.29) λ n λ n u(n, x) T T ± l l ( ) (4.26) T f(n, x) d + g(n, x) f(n, x) d + g(n, x) T u(n, x) + c(n)t u(n, x) = 0 (4.30) (4.27) n n l f(n, x) d + g(n, x) f(n, x) d + g(n, x) u(n l, x) + c(n)u(n l, x) = 0 (4.3) T u(n, x) u(n l, x) (4.32) T u(n, x) u(n ± l, x) 4.5.2 Lu = p 0 (x) d2 u + p 2 (x) du + p 2(x)u = 0 ( Lu = f d ) ( + g f d + g ) u + cu ( ) ( Lu = ff d2 u 2 + df du fg + gf + f + gg + ff = p 0 fg + gf + gg + f df = p f dg = p 2 c ) dg f u f f = p 0 /f 2 3 g = { p p ( 0 g + df )} f f 3 g Riccati G = g/f ( dg g2 f + p ) df g = f (p 2 c) (4.33) p 0 f p 0 dg G2 + p p 0 G = p 2 c p 0 (4.34) 3 38

4.5.3 {P n (x)} (4.5) x 2 (4.25) ( x 2 ) 2 d2 u 2 2x( x2 ) du + n(n + )( x2 )u = 0 (4.35) p 0 = ( x 2 ) 2, p = 2x( x 2 ), p 2 = n(n + )( x 2 ) (4.26) (4.27) f(n, x) = f(n, x) = x 2 g(n, x) (4.33) g(n, x) ( x 2 )g g 2 = n(n + )( x 2 ) c x 2 g g = ax + b b = 0, a = n, c = n 2 b 2 = 0, a 2 = (n + ), c 2 = (n + ) 2 g = nx, g 2 = (n + )x 2 c 2 (n) = c (n + ) g g 2 (4.26) (4.27) g = g(n, x), g 2 = g(n +, x) (4.35) = ( x 2 ) d nx ( x 2 ) d + (n + )x ( x 2 ) d + nx u + n 2 u (4.36) ( x 2 ) d (n + )x u + (n + ) 2 u (4.37) ( x 2 ) d + nx P n (x) = λ n P n (x) (4.38) ( x 2 ) d (n + )x P n (x) = λ n P n+ (x) (4.39) λ n λ n P n (x) = (2n)! 2 n n!n! xn (2n 2)! 2 n (n )!(n 2)! xn 2 +... P n (x) = (2n 2)! 2 n (n )!(n )! xn +... P n+ (x) = (2n + 2)! 2 n+ (n + )!(n + )! xn+ +... (4.40) (4.38) x n λ n = n (4.39) x n + λ n = (n + ) 39

( x 2 ) d + nx P n (x) = np n (x) (4.4) ( x 2 ) d (n + )x P n (x) = (n + )P n+ (x) (4.42) (4.4) (4.42) 3 (3.45) 4.5.4 {T n (x)} (4.7) x 2 ( x 2 ) 2 d2 u 2 x( x2 ) du + n2 ( x 2 )u = 0 = (n )x ( x 2 ) d ( x 2 ) d + (n + )x ( x 2 ) d + nx u + n(n )u (4.43) ( x 2 ) d nx u + n(n + )u (4.44) ( x 2 ) d nx T n (x) = nt n+ (x) (4.45) ( x 2 ) d + nx T n (x) = nt n (x) (4.46) (4.45) (4.46) 3 (3.56) 4.5.5 {S α n (x)} (4.9) x x 2 d2 u + (α + x)du 2 + nxu = 0 = x d x d n x + n + α x d n x d x + n + + α u n(n + α)u (4.47) u (n + )(n + + α)u (4.48) 40

x d n Sn α (x) = (n + α)sn (x) α (4.49) Sn α (x) = (n + )Sn+(x) α (4.50) x d x + n + α + (4.49) (4.50) 3 (3.65) 4.5.6 {H n (x)} (4.23) ( ) ( ) d d d d 2x + 2n u = 2x + 2(n + ) u = 0 (4.5) d H n(x) = 2nH n (x) (4.52) ( ) d 2x H n (x) = H n+ (x) (4.53) (4.52) (4.53) 3 (3.84) 4.6 4.6. P m n (x) = ( ) m ( x 2 ) m/2 dm P n (x) m (4.54) (3.42) m 0 m n Pn m (x) = ( )n+m 2 n ( x 2 m/2 dn+m ) n! n+m ( x2 ) n (4.55) Pn m m (n m)! (x) = ( ) (n + m)! P n m (x) (4.56) n m n Pn m (x)pl m (x) = (n + m)! 2 (n m)! 2n + δ nl (4.57) 4

(n m + )P m n+(x) (2n + )xp m n (x) + (n + m)p m n (x) = 0 (4.58) (cos θ + i sin θ cos ϕ) n = P n (cos θ) + 2 (4.5) n m= n! (n + m)! cos ϕp m n (cos θ) (4.59) } ( x 2 ) d2 u {n(n 2 2xdu + + ) m2 x 2 u = 0 (4.60) x = cos θ d = d sin θ dθ ( d sin θ d ) m2 sin θ dθ dθ sin 2 + n(n + ) u = 0 (4.6) θ P m n (cos θ) 4.6.2 Yl m (θ, ϕ) = ( ) (m+ m ) 2 2l + 4π /2 (l m )! P m l (cos θ)e imϕ (4.62) (l + m )! ( d sin θ d ) + d 2 sin θ dθ dθ sin 2 + n(n + ) u = 0 (4.63) θ dϕ2 2 ( = d2 2 + d2 dy 2 + d2 dz 2 ) ( u = 0) ( u = k 2 u) 2π π 0 0 Y m l (θ, ϕ)y m l (θ, ϕ)dθdϕ = δ ll δ mm (4.64) l l = ir ( ħ l = iħr ) l x = i(y d dz z d dy ) (4.65) l y = i(z d x d dz ) (4.66) l z = i(x d dy y d ) (4.67) 42

l 2 = sin θ ( ) d dθ sin θ d dθ + sin 2 θ d 2 dϕ l 2 x = i ϕ l 2 Yl m (θ, ϕ) = l(l + )Yl m (θ, ϕ) (4.68) l z Yl m (θ, ϕ) = myl m (θ, ϕ) (4.69) ( l ± = l x ± il y = e ±iϕ ± θ + i cot θ ) ϕ l ± m (4.70) l ± Y m l (θ, ϕ) = l(l + ) m(m ± )Y m+ l (θ, ϕ) (4.7) Y0 0 =, 4π 3 Y 0 = Y 0 2 = ± 3 cos θ, Y = 4π 4π sin θe±iϕ, 5 5 6π (3 cos2 θ ), Y 2 ± = 32π sin θ cos θe±iϕ, Y 2 ±2 = 5 32π sin2 θe ±2iϕ Y lm l=0 2 3... s p d f l = 2 l = 2 { Y lm = 2 ( ) m Yl m + Y m } { l Yl m = i 2 ( ) m Yl m Y m } l l 3 Y 00 s 4π 4π 3 3 Y sin θ sin ϕ 4π 4π y p y 3 3 Y 0 4π cos θ 4π z p z 3 3 Y sin θ cos ϕ 4π 4π x p x 5 5 Y 2 2 6π sin2 θ sin 2ϕ 4π xy d xy 5 5 Y 2 sin θ cos θ sin ϕ 4π 4π yz d yz 5 5 Y 20 6π (3 cos2 θ ) 6π (3z2 ) 5 5 Y 2 sin θ cos θ cos ϕ 4π 4π zx d zx Y 22 5 6π sin2 θ cos 2ϕ 43 d 3z2 5 6π (x2 y 2 ) d x 2 y 2

Y 00 Y Y 0 Y Y 2 2 Y 2 Y 20 Y 2 Y 22 44

A cos nπ = ( ) n, sin nπ = 0 (n : ) sin α sin β = {cos(α + β) cos(α β)} 2 sin α cos β = {sin(α + β) + sin(α β)} 2 cos α cos β = {cos(α + β) + cos(α β)} 2 e ix = cos x + i sin x (x n ) = nx n, x n = xn+ + C (n ) n + (log x) = x, = log x + C x (sin ax) cos ax = a cos ax, sin ax = + C a (cos ax) sin ax = a sin ax, cos ax = + C a (e ax ) = ae ax, e ax = eax a + C π e ax2 = a f(x)g (x) = f(x)g(x) f (x)g(x) n D n f(x)g(x) = nc k D n k f(x)d k g(x) D = d k=0 2 n (a + b) n = nc k a n k b k k=0 45