II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

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1 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh n 1 x), H n x) H nx) xh nx) + nh n x) = [5] H m x)h n x)e x dx = π n n! δ mn

2 II No. [1]P n x) P n x) = [n/] r= 1) r n r)! n r!n r)!n r)! xn r []P n x) n,, P n x) = 1) n P n x), P n 1) = 1 [3] P n x) = 1 d n n n! dx n x 1) n [4] n + 1)xP n x) = n + 1)P n+1 x) + np n 1 x) [ 1 x ) d ] n + 1)x P n x) = n + 1)P n+1 x) dx [ 1 x ) d ] dx + nx P n x) = np n 1 x), P n x) 1 x )P n x) xp nx) + nn + 1)P n x) = 1 [5] P m x)p n x) dx = 1 n + 1 δ mn

3 II No.3 [1]Pl m x) = 1) l+m Pl m x), Pl x) = P l x) [] Pl m x) = 1 x m/ dm ) dx P lx), m m [3] P m m l m)! l x) = 1) l + m)! P l m x) [4] [ 1 x ) d ] dx + mx ] [ 1 x ) d dx mx P m l x) = 1 x P m+1 x) P m l l x) = l + m)l m + 1) 1 x P m 1 x), Pl m x) [1 x ) d dx x d + ll + 1) m dx 1 x ] P m l x) = l 1 [5] 1 P m l x)pl m l + m)! x) dx = l + 1 l m)! δ ll

4 II No.4 [1]Y l m θ, φ) = 1) m Y lm θ, φ), Y lm π θ, φ + π) = 1) l Y lm θ, φ) [] e iφ θ + i cot θ ) Y lm θ, φ) = l m)l + m + 1)Y lm+1 θ, φ) φ e iφ θ + i cot θ ) Y lm θ, φ) = l + m)l m + 1)Y lm 1 θ, φ) φ, Y lm θ, φ) [ 1 sin θ ) sin θ θ θ + 1 sin θ ] + ll + 1) Y φ lm θ, φ) = π π [3] sin θ dθ dφ Y lm θ, φ) Y l m θ, φ) = δ ll δ mm [4] Y lm θ, φ) = 1) m+ m )/ l + 1 l m )! 4π l + m )! P m l cos θ)e imφ [5] r, θ, φ) f = 1 r f ) + 1 sin θ f ) 1 f + r r r r sin θ θ θ r sin θ φ

5 II No.5 [1]Y lm m 1 Y lm = 1 [ 1) m Y lm + Y l m ] m 1 Y lm = 1 i [Y lm 1) m Y l m ] []Y lm m 1 Y lm = m 1 Y lm = l + 1 l m )! π l + m )! P l m cos mφ l + 1 l m )! π l + m )! P m l sin m φ π π [3] sin θ dθ dφ Y lm Y l m = δ ll δ mm [4]l = 1 p ) 3 x Y 11 = 4π r Y 1 1 = 3 y 4π r Y 1 = 3 z 4π r, 3 3 P Y 11 Y 1 1 Y 1 ) = Y 11 Y 1 1 Y 1 )P, P [5]l = d ) 15 x y Y = Y 16π r = 15 zx Y 1 = Y 4π r 1 = 15 xy 4π r 15 yz Y 4π r = 5 3z r 16π r, 5 5 P Y Y Y 1 Y 1 Y ) = Y Y Y 1 Y 1 Y )P, P

6 II No.6 [1]L n x) L n x) = n r= 1) r n!) r!) n r)! xr []L n ) = n! [3] L n x) = e x dn dx n x n e x) [4] L n+1 x) = n + 1 x)l n x) n L n 1 x) x d ) dx + n + 1 x L n x) = L n+1 x) x d ) dx n L n x) = n L n 1 x), L n x) xl nx) + 1 x)l nx) + nl n x) = [5] L m x)l n x)e x dx = n!) δ mn

7 II No.7 [1]L k nx) 1)k exp xt ) = 1 t) k+1 1 t n=k L k nx) tn k n! n k []L k nx) L k nx) = 1) r+k n!) r + k)!n r k)!r! xr r= [3] L k nx) = 1) k n! n k)! x k e x dn k dx n k x n e x) [4] 1 k ) L k n + 1 n+1x) + x + k n 1)L k nx) + n L k n 1x) = x d ) dx x + n + 1 L k nx) = 1 k ) L k n + 1 n+1x) x d ) dx n + k L k nx) = n L k n 1x), L k nx) xl k n x) + k + 1 x)l k n x) + n k)l k nx) = [5] L k mx)l k nx)x k e x dx = n!)3 n k)! δ mn

8 II No.8 [1]Re z >, Γz) Γz + 1) = zγz) []Γ1) = 1,, Γn + 1) = n! [3] e x dx = π [4]Γ ) 1 = π,, Γ n + 1 ) = n 1)!! n π [5]Γz + 1) = zγz) Γ1) = 1, Γz) z = m m =, 1,, ) 1)m m!

9 II No.9 [1]Re z >, n 1 t n) n t z 1 dt = n! n z zz + 1) z + n) [] 1 Γz) = zeγz m=1 [ 1 + z ) ] e z m m [3] Γz)Γ1 z) = π sin πz n+1 n!n 1 [4], lim n n + 1)!! = π [5] πγz) = z 1 Γz)Γ z + 1 )

10 II No.1 [1]ft) = t xlog t t = x ft) t x) x xlog x +, x Γx + 1) πxx x e x π []Re p >, Re q > Bp, q) = cos p 1 θ sin q 1 θdθ [3]Γp)Γq) = { } { e u u p 1 du } e v v p 1 dv u = r cos θ, v = r sin θ, Γp + q)bp, q) [4] ψz + 1) = 1 z + ψz) [5]ψz) = 1 z γ 1 z + m 1 ), ψ1) = γ,, m m=1 ψn + 1) = 1 n + 1 n γ

11 II No.11 [1] e z t 1 t ) t n= J n z)t n, J n z) = s= 1) s z ) n+s s!s + n)!, J n z) = 1) n J n z) J n z) = 1) n J n z) []J z) + J m z) = 1 m=1 [3] J n z) = 1 π π π e iz sin θ inθ dθ = 1 π π cosz sin θ nθ)dθ [4]J ν z), d dz ν ) J ν z) = J ν+1 z) z d dz + ν ) J ν z) = J ν 1 z) z [5]J ν z) y + 1 ) z y + 1 ν y = z

12 II No.1 [1]N ν z) J ν z), d dz ν ) N ν z) = N ν+1 z) z d dz + ν ) N ν z) = N ν 1 z) z []J ν z) cos πν N ν z) sin πν = J ν z) J ν z) sin πν +N ν z) cos πν = N ν z),, ν = n ) 1) n J n z) = J n z) 1) n N n z) = N n z) [3]H ν 1) z) = e iπν J ν z) + J ν z) i sin πν H ν ) z) = eiπν J ν z) J ν z) i sin πν [4]H νz) 1) = e iπν H ν 1) z) H ) ν z) = e iπν H ν ) z) [5] d { z ν J ν z) } ) n 1 = z ν J ν+1 z) J ν+n z) = 1) n z ν+n d { z ν J ν z) } dz z dz, ) n 1 J n z) = 1) n z n d J z) z dz ) n J n+ 1 z) = 1) n z n+ 1 1 d { } z 1 J 1 z) z dz

13 II No.13 d [1]I ν z) J ν z), dz ν ) I ν z) = I ν+1 z) z d dz + ν ) I ν z) = I ν 1 z), I ν z) y + z 1 + ν 1 z y z ) y = d []K ν z) I ν z), dz ν ) K ν z) = K ν+1 z) z d dz + ν ) K ν z) = K ν 1 z), K ν z) z y + 1 ) z y 1 + ν y = z d [3]j n z) J n+ 1 z), dz n ) j n z) = j n+1 z) z d dz + n + 1 ) j n z) = j n 1 z), j n z) z y + z y + 1 nn + 1) z ) y = [4]j n z) = 1) n z n 1 z ) n d j z) dz [5] r, θ, φ) ur, θ, φ) = fr)gθ)hφ), fr), gθ), hφ)

14 II No.14 [1] x ) x m= ρ x) N, k= T/ x) ),, y ) x P = l, ), Q = l, ), I[y] y x), y x) I[y ], ρ T N, x [], x 1, x Lt, x 1, ẋ 1, x, ẋ ), S[x 1, x ] = t t 1 Lt, x 1, ẋ 1, x, ẋ ) dt, L, T V L = T V,, x i t 1 ), x i t ) i = 1, ) d ) L L = i = 1, ) dt ẋ i x i [3][1], T l, ρ,,. x, t yx, t), L = S[y] =, y t = T ρ l l [ t t 1 ρ ) y T t ) ] y dx x L dt δs[y] = y x [4] l = sinh 1 =.35) P = 1, ), Q = 1, ), I[y] = I[y] y x) Q P y ds [5][1], y a x) = a cos πx l, a, I[y a ] a Ia), Ia) a a, a Ia ), Ia ) [1]

15 II No.15 1 α 1 [1] δx) = lim δx) = lim e x α π x α + α α πα,, δx)dx = 1 []c, δcx) = 1 δx) c [3] b a fx )δ x x)dx = f x) [4]δx) Hx) Hξ x) = ξ a δx x )dx [5] [, 1], x D N x) = N sin nπx sin nπx N = 1, 1, 1 x < x < 1 ). n=1

16 II No.16 [1] f = b fx) x dx, x x = δx x ) f g = b a a fx) gx)dx []ˆx = b x x x dx, ˆx n = b a a x n x x dx [3] x ˆx n = x n x, Aˆx) =. a nˆx n, x Aˆx) = Ax) x n= [4]ˆk =. L/ L/ x i d ) x dx, dx ˆk L/ n = x i d ) n x dx L/ dx [5] x ˆk n = A i d dx i d ) n x, Aˆk) = dx ) x a nˆkn, x Aˆk) = n=

17 II No.17 [1]δk k ) = 1 e ik k )x dx δx x ) = 1 e ikx x ) dk π π [] Ûa = expiˆka) Ûa = x x + a dx, Û a x a [3]δx) = 1 e ikx dk, δ m) x) = 1 π π δ m) x) δx) m ik) m e ikx dk, [4]ˆx = k i d ) k dk,, k ˆx = i d k dk dk [5] x x n fx) =, fx) = c δx)+ +c n 1 δ n 1) x), c m m =,, n 1)

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin

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