(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :. (Poisson). ρ = V V =.. t (x, y, z) T (x, y, z, t) T T = κ T κ t.. T/ t = T =.... i

1 1 1.1......................... 1 1........................... 5 1.3............................... 7 1.4........................... 9 1.5.......................... 1 1.6............................ 13 1.7........................... 15 1.8..................... 17 1.9....................... 19 1.1...................... 1 1.11...................... 5 1.1....................... 7 1.13.................................... 8 1.14.......................... 31 37.1............................ 37.......................... 41.3....................... 43.4.......................... 45.5.................... 48.6.......................... 5.7 z < 1.................... 53.8........................... 59.9....................... 63.1....................... 66.11 ν P ν (cos θ) =...................... 67 3 7 3.1 L p (Ω).................................... 7 3................................ 74 3.3........................... 76 3.4.................................... 81 ii

3.5............................ 83 3.6.......................... 85 4 9 4.1............................ 9 4....................... 9 4.3.................. 93 4.4................................ 96 4.5..................... 98 4.6.................. 1 4.7....................... 1 4.8.......................... 11 4.9...................... 13 4.1................. 16 5 11 5.1............................ 11 5............................... 111 5.3........................... 11 5.4............................ 113 5.5......................... 114 5.6.................... 115 5.7........................... 116 5.8...................... 118 5.9 z..................... 11 5.1....................... 14 5.11............................... 16 A 19 A.1............................ 19 A....................... 13 A.3......................... 134 A.4........................... 137 A.5......................... 139 iii

1 A 1.1 z d f dz + 1 df z dz + ( 1 ν ) f = ν : (1.1) z (Bessel), z = f(z) = z α m= a m z m a a, a 1,... α m= {(α + m)(α + m 1) + (α + m) ν } a m z α+m + a m z α+m = z z α α ν =, α = ±ν z α 1 a 1 =, z α+m {(α + m) ν }a m + a m = a 1 = a 3 = a 5 = =, α = ν m m Γ(z) a m = ( 1)m Γ(ν + 1) a m m! Γ(m + ν + 1) m=, a = { ν Γ(ν + 1)} 1 a m = ( 1) m m+ν m! Γ(m + ν + 1) 1

, ( z ν ( 1) f(z) = J ν (z) ) m m! Γ(ν + m + 1) m= ( z ) m (1.) ν z (z/) ν J ν (z) (D Alembert) α = ν J ν (z) ν ν, J ν (z) ν J ν (z) J ν (z), ν = ±1/ ( 1) m ( z 1/+m J 1/ (z) = m! Γ(m + 1/ + 1) ) m= z ) = (1 z π 3! + z4 z 5! sin z = (1.3) π z ) J 1/ (z) = (1 z πz! + z4 4! cos z = (1.4) πz z J 1/ (x) 1 J 1/ (x) 3π π π 4π 1 1.1:, ν n J n (z) J n (z) 1 lim ν n Γ(ν + m + 1) = n + m + 1 J n (z) = lim ν n J ν(z) = ( z ) n m=n ( 1) m z ) m m! Γ(m n + 1)(

( z n ( 1) = ) n ( 1) k z ) k k! Γ(n + k + 1)( k= J n (z) = ( 1) n J n (z) n : ν J ν (z) J ν (z), ν J n (z) J n (z), J n (z) Y ν (z) = J ν(z) cos νπ J ν (z) sin νπ (ν ) (1.5) Y ν (z) J ν (z) Y ν (z) ν (Neumann) ν n (1.5) ν n J ν (z) cos νπ J ν (z) Y n (z) = lim (1.6) ν n sin νπ Y n (z),, Y n (z) J n (z) (1.6) ν Y n (z) = 1 { π sin nπ J n (z) + cos nπ [ J ν (z) π cos nπ ν [ J ν (z) ν ]ν=n ( 1)n π = 1 π ]ν=n [ J ν (z)] } ν ν=n [ J ν (z)] (1.7) ν ν=n J ν (z) ν = ( z ) ν z log m= + ( z ) ν ( 1) m (z ) m m! Γ(ν + m + 1) m= ( 1) m m! [ ν 1 Γ(ν + m + 1) ] ( z ) m, (1.1) ψ(z) = Γ (z) Γ(z) = γ 1 z + z 1 m(m + z), m=1 γ : (Euler) [ J ν (z) ν ]ν=n = m= ( 1) m { m! Γ(n + m + 1) log z ψ(n + m + 1) } ( z ) n+m 3

(1.7), ν < n J ν (z) = n 1 m= ( 1) m m! Γ( ν + m + 1) (z ) ν+m ( 1) m + m! Γ( ν + m + 1) m=n (z ) ν+m ν,, (1.7) 1 Γ(ν m) sin(ν m)π = Γ( ν + m + 1) π, m < n [ J ν (z)] ν [ { ( z ) ν+m Γ(ν m) sin(ν m)π }] ν π ν=n = 1 [ ( z ) ν+mγ(ν { m) ψ(ν m) sin(ν m)π π + π cos(ν m)π log z sin(ν m)π}] ν=n = ( z ) n+mγ(n m) cos(n m)π ν=n = n 1 = ( 1) m ( 1) n m Γ(n m) m! m= ( 1) m { + m! (m n)! n 1 m= m=n ( 1) n Γ(n m) m! + k= (z ) n+m ( 1) k+n { (k + n)! k! (z ) n+m log z + ψ(m n + 1) } ( z log z + ψ(k + 1) } ( z ) n+k n Y n (z) = π J n(z) log z 1 π 1 π m= n 1 m= (n m 1)! m! ψ(m + 1) + ψ(n + m + 1) m! (n + m)! (z ) n+m ) n+m ( 1) m( z ) n+m (1.8) Y n (z) (1.5) ν = ±1/ Y 1/ (z) = πz cos z Y 1/(z) = sin z (1.9) πz 4

1 Y 1/ (x) 3π π Y 1/ (x) π 4π 1 1.: 1. ν n 1.1 J n (z) J n (z) = 1 π π π e iz cos t e in(t π/) dt n : (1.1) Hanse f(z) = 1 π e iz cos t e in(t π/) dt n : (1.11) π π e iz cos t (iz cos t) m = m! m= = m= (iz) m m! ( e it + e it ) m (1) m (iz) m = m! m= m mc r e irt e i(m r)t r= f(z) = 1 π (1) m (iz) m m! m= m π mc r e in(t π/) e irt e i(m r)t dt r= π { π n + r m e it(n+r m+r) dt = π π n + r m = 5

r r = (m n)/ (1) m (iz) m f(z) = e inπ/ mc (m n)/ m! m= (m n)/ = p mc (m n)/ = p+n C p = (p + n)! p! (p + n)! f(z) = p= (z ) p+n (i) p (p + n)! (p + n)! p! (p + n)! = ( z ) n m= ( 1) m m! (m + n)! (z ) m (1.) f(z) = J n (z) 1.1 1. J ν (z) J ν (z) = 1 z ) ν 1 Γ(ν + 1/) Γ(1/)( t ν 1/ (1 t) 1/ cos[z(1 t) 1/ ] dt Re(ν + 1/) > (1.1) (1.) ν + 1/ > ( z ν ( 1) ) m z m ( z ) ν ( 1) = m! Γ(ν + m + 1)( ) m Γ(m + 1/) Γ(ν + m + 1) Γ(1/) Γ(m + 1) zm ( z ν ( 1) = ) m z m B(ν + 1/, m + 1/) Γ(ν + 1/) Γ(1/) (m)! ( z ν ( 1) = ) m z m 1 t ν 1/ (1 t) m 1/ dt Γ(ν + 1/) Γ(1/) (m)! t ν 1/ ( 1)m z m (1 t) m 1/ (m)! m= (Lebesgue) (, 1), (, 1), J ν (z) ( z ) ν 1 J ν (z) = Γ(ν + 1/) Γ(1/) 1 m= ( 1) m (m)! zm t ν 1/ (1 t) m 1/ dt Re(ν + 1/) > 6

m= J ν (z) = ( 1) m (m)! zm (1 t) m 1/ = (1 t) 1/ cos[z(1 t) 1/ ] 1 z ) ν 1 Γ(ν + 1/) Γ(1/)( t ν 1/ (1 t) 1/ cos[z(1 t) 1/ ] dt Re(ν + 1/) > t = sin θ J ν (z) = z ) ν π/ Γ(ν + 1/) Γ(1/)( 1 cos(z cos θ) sin ν θ dθ, Re(ν + 1/) > (1.13) (Poisson),1 t = ξ t ν 1/ (1 t) 1/ cos[z(1 t) 1/ ] dt = 1 1 (1 ξ ) ν 1/ cos(zξ) dξ 1 z ) ν 1 J ν (z) = (1 ξ Γ(ν + 1/) Γ(1/)( ) ν 1/ e izξ dξ, Re(ν + 1/) > (1.14) 1.3 (1.11) f(x, y) x, y, x, y f(x, y) dx, f(x, y) dy, y f(x, y) f(x, y) = 1 f(ξ, y) e iω(x ξ) dξ dω π ξ f(ξ, y) = 1 π f(ξ, η) e iω (y η) dη dω 7

(p83) f(x, y) = 1 (π) f(ξ, η) e iω(x ξ)+iω (y η) dω dω dξ dη x = r cos φ, ξ = ρ cos ψ, ω = σ cos α y = r sin φ, η = ρ sin ψ, ω = σ sin α ωx + ω y = σr cos(α φ), ωξ + ω η = σρ cos(ψ α) dξ dη = ρ dρ dψ, dω dω = σ dσ dα, f(x, y) = g(r)e inφ n : g(r)e inφ = 1 f(ρ cos ψ, ρ cos ψ) e iσr cos(α φ) iσρ cos(ψ α) ρ σ dρ dψ dσ dα (π) { 1 π = σ dσ g(ρ) ρ dρ e iσr cos(α φ) dα 1 π } e inψ e iσρ cos(ψ α) dψ π π (ψ α = t ) = = = σ dσ { 1 π σ dσ σ dσ π π = e inφ σ dσ g(ρ) ρ dρ π π ( 1 )} e iσr cos(α φ) dα e inπ/ e inα e i( σρ) cos t e in(t π/) dt π π { 1 π } g(ρ) e iσr cos(α φ) dα e inα e inπ/ ( 1) n J n (σρ) ρ dρ π π ( ) g(ρ) e inπ/ e inφ J n (σr) e inπ/ ( 1) n J n (σρ) ρ dρ g(ρ) J n (σr) J n (σρ) ρ dρ g(r) = σ dσ g(r) = h(σ) J n (σr) σ dσ h(σ) = π g(ρ) J n (σr) J n (σρ) ρ dρ (1.15) g(ρ) J n (σρ) ρ dρ (1.16), h(σ) g(r) n n ν Re(ν) > 1 8

1.4 1.3 J ν 1 (z) + J ν+1 (z) = ν z J ν(z) (1.17) (1.) J ν 1 (z) + J ν+1 (z) = = 1 Γ(ν)( z ) ν 1 + m=1 ( 1) m z ) ν+m 1 ( 1) + m! Γ(ν + m)( m z ) ν+m+1 m! Γ(ν + m + )( m= { ( 1) m 1 m! Γ(ν + m) 1 z ) ν+m 1 (m 1)! Γ(ν + m + 1)}( ν( 1) m z ) ν+m 1 m! Γ(ν + m + 1)( m= = Γ(ν)( 1 z ) ν 1 + m=1 = ν z m= ( 1) m m! Γ(ν + m + 1)( z ) ν+m = ν z J ν(z) J ν 1 (z) + J ν+1 (z) = ν z J ν(z) 1.4 J ν 1 (z) J ν+1 (z) = J ν (z) (1.18) J ν (z) z dj ν (z) dz = 1 = 1 ( 1) m (ν + m) z ) m+ν 1 m! Γ(ν + m + 1)( ( 1) m z ) m+ν 1 1 ( 1) + m! Γ(ν + m)( m z ) m+ν 1 (m 1)! Γ(ν + m + 1)( m= m= = 1 J ν 1(z) 1 J ν+1(z) m=1 J ν 1 (z) J ν+1 (z) = J ν (z) 9

(1.17), (1.18) { zjν (z) + νj ν (z) = zj ν 1 (z) zj ν (z) νj ν (z) = zj ν+1 (z) (1.19) 1.5 Y ν 1 (z) + Y ν+1 (z) = ν z Y ν(z) (1.) Y ν 1 (z) + Y ν+1 (z) = J ν 1(z) cos(ν 1)π J ν+1 (z) + J ν+1(z) cos(ν + 1)π J ν 1 (z) sin(ν 1)π sin(ν + 1)π = 1 [{ } ] J ν 1 (z) + J ν+1 (z) cos νπ + J ν+1 (z) + J ν 1 (z) sin νπ = 1 [ ν { }] J ν (z) cos νπ J ν (z) = ν sin νπ z z Y ν(z) Y ν 1 (z) + Y ν+1 (z) = ν z Y ν(z) Y ν 1 (z) Y ν+1 (z) = Y ν (z) zy ν (z) + νy ν (z) = zy ν 1 (z) (1.1) zy ν (z) νy ν (z) = zy ν+1 (z) 1.5 d u dz + p(z)du + q(z)u = (1.) dz 1

{ u(z) = U(z) exp 1 } p(z)dz d U dz + G(z) U = (1.3) G(z) = q(z) 1 dp dz 1 4 p (z) (1.3) U V (1.) V v U, V d U dz V U d V dz = V du dz U dv dz = c ( ) (1.4) U, V (Wronsky) U, V c = U(z) = c V (z) (c : ), U, V, (1.4) U(z) U d ( V ) = c dz U V U = c dz U (z) + k (k : ), U, V V = cu U dz + k U c u, v v(z) = c u(z) u (z) exp[ p(z)dz] dz + k u(z) 11

(Sturm). 1.1 ( (Sturm) ) x u, v d u d v + h(x)u =, + g(x)v = (1.5) dx dx h(x), g(x) x u(x) x = a, x = b a x b g(x) h(x) [a, b] v(x) [a, b] g(x) h(x) a, b v (a, b) v (1.5) u(a) = u(b) = [ v du ] x=b = dx x=a d [ v du ] dx dx udv + uv [ h(x) g(x) ] = dx b a [ g(x) h(x) ] uvdx (1.6) u(x) > a < x < b, u (a) > > u (b) g(x) h(x) (a x b) v(x) [a, b], v(x) >, a x b = v(b)u (b) v(a)u (a) < g(x) h(x) a, b v (a, b) v v(x) > (1.6) v(a) = v(b) = v(a) = v(b) > v(b) = v(a) > (1.5) d dx [ du] d p(x) + h(x) u =, dx dx, d [ du p(x) v dx dx [ p(x) v du dx [ dv ] p(x) + g(x) v = dx dv ] [ ] p(x) u + h(x) g(x) u v = dx ] x=b x=a = b a [ g(x) h(x) ] u v dx p(x) a x b (1.1) z 1

x d ( x df ) ) + (x ν f = dx dx x, p(x) = x (, ) J ν (x) (, ) Y ν (x) J ν (x) Y ν (x) Y ν (x) J ν (x) 1.6 ν J ν (x) 1.6 J ν (x) [, ) α, β J ν (αx) J ν (βx), d J ν (αx) dx + 1 x dj ν (αx) dx + (α ν x )J ν(αx) = d J ν (βx) + 1 dj ν (βx) + (β ν dx x dx x )J ν(βx) = d ( x dj ν(αx) J ν (βx) x dj ) ν(βx) J ν (αx) = (β α ) x J ν (αx) J ν (βx) dx dx dx x b ν > 1 b [ (β α ) x J ν (αx) J ν (βx) dx = x dj ν(αx) J ν (βx) x dj ν(βx) dx dx ] x=b J ν (αx) x= (1.7) x = α J ν (αb) = β α J ν (x) J ν (βb) = = (β α ) b J ν (αx) x dx α ν > 1 J ν (z) J ν (z) [, ) U(x) J ν (x) x, D(x) 1 + 1/4 ν x 13

d U dx + D(x) U = V (x) cos ( 1 x) d V dx + 1 V = x D(x) > 1 V (x) [, ) U(x) J ν (x) Y ν (x) Y (z) J ν (z) 1.7 Y (z) Re(z) > ν = Y (z) z = Y (z) = π (log z ) + γ + O(z ) z = zy (z) = /π + o(z) (β α ) b x Y (αx) Y (βx) dx = [ x Y (βx) dy (αx) dx x Y (αx) dy ] (βx) 4 dx x=b π log β α (1.8) Y (z) βb αb β = Ae iθ, α = Ae iθ, A, θ > b Y (αx) 4θ x dx = A π sin θ < θ < π/ Y (z) Re(z) > 14

1.7 1.8 α k (k =, 1,,... ) J n (αb) = (n :, b > ) (1.9) b b r J n (α k r)j n (α l r) dr = α k α l (1.3) r Jn(α k r) dr = b [ Jn (α k b) ] (1.31) n α, β J n (αr), J n (βr) 1.6 (1.7) r = b { α J n (αb)j n (βb) β J n (βb)j n (αb) } = (β α ) α, β J n (αb) = α, α 1, α,... b r J n (αr)j n (βr) dr (1.3) b r J n (α k r)j n (α l r) dr = α k α l J n (α r), J n (α 1 r), J n (α r),... α = α k, β = α k + ε, ε 1, J n (βb) = J n (bα k ) + J n (bα k ) bε + ( ε ) (1.3) b α k ε [ J n (α k b)] + = αk ε b r J n (α k r)j n (α k r + εr) dr 15

ε b r Jn(α k r) dr = b [ Jn (α k b) ] (1.33) α (1.9) α J n (αb) + h J n (αb) = (1.34) (1.3) 1.14 1.9 α k (k =, 1,,... ) (1.34) b r J n (α k r)j n (α l r) dr = α k α l b r J n (α k r) dr = 1 (1.35) α (h b + α k b n ) J n (α k b) k (1.9) (1.34) α, α 1, α,... 1.8 b r J n (α k r)j n (α l r) dr = α k α l β = α k + ε (1.3) b { α k J n (α k b)j n (βb) (α k + ε)j n (βb)j n (α k b) } [ = b α k J n = b h ε α k (α k b) { J n (α k b) + J n (α k b) bε + } (α k + ε)j n (α k b) { J n (α k b) + J n (α k b) bε + }] J n (α k b) bε { J n (α k b) + bα k J n (α k b) } J n (α k b) + = ε {b h + (α k b n )} α k J n (α k b) + ε b r J n (α k r) dr = 1 α k (h b + α k b n ) J n (α k b) 16

1.8 H (1) ν (z), H () ν (z), { Hν (1) (z) = J ν (z) + iy ν (z) H ν () (z) = J ν (z) iy ν (z) (1.36) J ν (z) = 1 { } H (1) ν (z) + H () ν (z) Y ν (z) = 1 { } H (1) ν (z) H () ν (z) i H (1) ν (z) = J ν (z) + i [ J ν (z) cos νπ J ν (z) ] sin νπ = J ν(z) e iνπ J ν (z) (1.37) i sin νπ H () ν (z) = eνπi J ν (z) J ν (z) i sin νπ, J ν (z), Y ν (z) (1.36) H (1) ν 1 (z) + H (1) ν+1 (z) = ν z H ν (1) (z) (1.38) H () ν 1 (z) + H () ν+1 (z) = ν z H ν () (z) H (1) ν 1 (z) H (1) (1) ν+1 (z) = H ν (z) H () ν 1 (z) H () () ν+1 (z) = H ν (z) (1.39) z iz d f dz + 1 z df ( ) dz 1 + ν f = (1.4) z J ν (z), Y ν (z) J ν (iz) J ν (iz) = e νπi/ ( z ) ν m= 1 z ) m m! Γ(ν + m + 1)( I ν (z) ( z ν 1 z ) m I ν (z) = (1.41) ) m! Γ(ν + m + 1)( m= 17

K ν (z) K ν (z) = π I ν (z) I ν (z) (1.4) sin νπ ν I ν (z) K ν (z), ν n ( ) I n (z) = I n (z) K n (z) = lim K ν (z) ν n, Y n (z) ( z ψ(m + 1) ( z ) m K (z) = log I (z) + (1.43) ) (m!) K n (z) = 1 n 1 m= + ( 1) n+1 ( 1) m (n m 1)! m! m= 1 m! (n + m)! ( z m= ) m n ( z n+m { ( z log ) ) 1 ψ(m + 1) 1 } ψ(n + m + 1) (1.44) J ν (iz) = e νπi/ I ν (z), I ν (z) = e νπi/ J ν ( iz) (1.45) } I ν 1 (z) + I ν+1 (z) = i e {J νπi/ ν 1 (iz) J ν+1 (iz) = i e νπi/ d d(iz) J ν(iz) I ν 1 (z) + I ν+1 (z) = I ν (z) I ν 1 (z) I ν+1 (z) = ν z I ν(z) K ν 1 (z) K ν+1 (z) = ν z K ν(z) K ν 1 (z) + K ν+1 (z) = K ν (z) H ν (1) (iz) = J ν(iz) e iνπ J ν (iz) i sin νπ = e νπi/ I ν (z) e νπi/ I ν (z) i sin νπ = πi e νπi/ K ν (z) 18

K ν (z) = (πi/) e νπi/ H ν (1) (iz), K ν (z) = K ν (z) K ν (z) = (πi/) e νπi/ H ν (1) (iz) (1.46), (1.13) Re(ν + 1/) > ( z ) ν I ν (z) = e νπi/ 1 1 J ν (iz) = (1 ξ ) ν 1/ e zξ dξ π Γ(ν + 1/) 1 1 ( z ) ν { 1 1 } = (1 ξ ) ν 1/ e zξ dξ + (1 ξ ) ν 1/ e zξ dξ π Γ(ν + 1/) ( z ) ν 1 = (1 ξ ) ν 1/ cosh(zξ) dξ π Γ(ν + 1/) ( z ) ν 1 I ν (z) = (1 ξ ) ν 1/ cosh(zξ) dξ, Re(ν + 1/) > (1.47) π Γ(ν + 1/) 1.9 1.1 sin νπ J ν (z)j ν 1 (z) J ν (z)j ν+1 (z) = πz J n (z)y n (z) J n (z)y n (z) = πz (1.48) (1.49) d ( z df ) + dz dz ) (z ν f = z, u, v d [ z (uv u v) ] = dz 19

u(z) v (z) u (z) v(z) = c z u(z), v(z) c, ν J ν J ν J ν (z)j ν (z) J ν (z)j ν (z) = c z (1.19) J ν (z) = ν z J ν(z) J ν+1 (z), J ν (z) = ν z J ν(z) + J ν 1 (z) J ν (z)j ν (z) J ν (z)j ν (z) = J ν (z)j ν 1 (z) J ν (z)j ν+1 (z) = c z, z 1 z 1 { } c = lim z J ν (z)j ν 1 (z) J ν (z)j ν+1 (z) z J ν (z) ( z ν { J ν (z) = ) 1 } Γ(ν + 1) +, z ( z 1 { 1 } J ν (z)j ν 1 (z) J ν (z)j ν+1 (z) = + ) Γ(ν + 1)Γ( ν) Γ(ν + 1)Γ( ν) = π/ sin π(ν + 1) sin νπ c = π J ν (z)j ν 1 (z) J ν (z)j ν+1 (z) = ν ν J ν 1 (z)j ν (z) J ν (z)j ν+1 (z) = sin νπ πz sin νπ πz (1.5) ν n, J n (z) Y n (z) (1.1) J n (z)y n (z) J n (z)y n (z) = πz

1.1 J ν (z) Re(ν) > / b w(z) = z ν e izt T (t) dt (1.51) a T (t) a b a, b d w dz = ν(ν 1) zν dw dz = νzν 1 b a b a b e izt T (t) dt + z ν it e izt T (t) dt a b b e izt T (t) dt + νz ν 1 it e izt T (t) dt z ν e izt t T (t) dt a d w dz + 1 z b = z ν a dw ( ) dz + 1 ν w z e izt (1 t ) T dt + z ν 1 (ν + 1)i a b a t e izt T dt b { = i z ν 1 (ν + 1) t T d [ (t 1)T ]} [ ] b e izt dt i z ν 1 (1 t ) T e izt dt a d [ (t 1) T ] = (ν + 1) t T (1.5) dt [ ] b (1 t ) T e izt = (1.53) a T (t) a b w(z) (1.5) T (t) = (t 1) ν 1/ (1.53) [ ] b [ ] b (t 1) T e izt = e izt (t 1) ν+1/ = a a a b t e izt (t 1) ν+1/ a b t = a, t = b e izt (t 1) ν+1/ a b (t 1) ν+1/ ν + 1/ t t = 1 t = 1 Riemann a 1

t 1 1 1.3: (1+,-1-) 1 1 i t 1 arg(t 1) = nπ S n S n 1 < t < 1 arg(t 1) = (n 1)π S n S n+1 S t = 1 t = 1 Riemann 1.3 t = 1, 1 e izt (t 1) ν+1/ t = 1 t 1 = e iθ, π < θ < π θ π π (t 1) ν+1/ π(ν + 1/) (t + 1) ν+1/ = ( + e iθ ) ν+1/ θ π π (t 1) ν+1/ π(ν + 1/) t = 1 t (t 1) ν+1/ π(ν + 1/) e izt e izt (t 1) ν+1/ (1+, 1 ) w(z) = z ν e izt (t 1) ν 1/ dt (1.54) a = b = C θ { 1 + e iθ ( π < θ π) C : [ π, 3π] θ t(θ) = 1 + e i(θ π) (π < θ 3π) t 1 = r(θ)e iϕ(θ) ϕ [ π, 3π] { θ + arg( + e iθ ) ( π < θ π) ϕ(θ) = (θ π) + arg( + e i(θ π) ) (π < θ 3π)

ϕ( π) = π, ϕ(3π) = π (1.54) 3π w(z) = z ν e izt(θ) r(θ) ν+1/ e iϕ(θ)(ν+1/) t (θ) dθ π [e izt(θ) r(θ) ν+1/ e iϕ(θ)(ν+1/) ] 3π J ν (z) π = 1.11 J ν (z) J ν (z) = Γ(1/ ν) ( z ) ν (1+, 1 ) πi e izt (t 1) ν 1/ dt, Re(ν) + 1/ > (1.55) π ν + 1/ (1.54) t = ±1 t 1.3 e izt i m z m t m = m! m= (1+, 1 ) e izt (t 1) ν 1/ dt = i m z m (1+, 1 ) t m (t 1) ν 1/ dt m! m= (t = ) ( 1) m z m (1+, ) = t m (t 1) ν 1/ dt (m)! m= (u = t ) ( 1) m z m (1+, ) = u m 1/ (u 1) ν 1/ du (m)! m= (1+, ) 1.4 Re(ν) > 1/ u =, u = 1 u u 1 = re iθ θ = π 3

T u u = u = 1 U 1.4: (1+,-) θ = π 1 u m 1/ (u 1) ν 1/ du = e iπ(ν+1/) (1 r) m 1/ r ν 1/ dr u m 1/ (u 1) ν 1/ du = e iπ(ν+1/) (1 r) m 1/ r ν 1/ dr (1+, ) ] u m 1/ (u 1) ν 1/ du = [e 1 iπ(ν+1/) e iπ(ν+1/) (1 r) m 1/ r ν 1/ dr = i sin π(ν + 1/) z ν (1+, 1 ) e izt (t 1) ν 1/ dt = i sin π(ν + 1/) Γ(ν + 1/) 1 ( 1) m z m+ν m= (m)! Γ(m + 1/)Γ(ν + 1/) Γ(m + ν + 1) Γ(m + 1/) Γ(m + ν + 1/) (1.4) Γ(m + 1/) π Γ(m) π = (m)! Γ(m) (m)! = m 1 m m! z ν (1+, 1 ) e izt (t 1) ν 1/ dt = i sin π(ν + 1/) Γ(ν + 1/) ( z ) ν π ν ( 1) m z ) m m! Γ(m + ν + 1)( m= = i π Γ(1/ ν) ν J ν (z) Γ(1/ ν) ( z ) ν (1+, 1 ) J ν (z) = πi e izt (t 1) ν 1/ dt, Re(ν) + 1/ > π ν + 1/ 4

1.11 a b (t 1) ν+1/ e izt = 1.1 J ν (z) Γ(1/ ν) ( z ) ν J ν (z) = πi e νπi e izt (t 1) ν 1/ dt, Re(z) > (1.56) π C ν + 1/ Re(z) > (t 1) ν+1/ e izt t i 1.5 C i 1 +1 i Re(z) > w(z) C S D 1 1 D 1 1 C L 1.5: C t > 1 (1.18) (t 1) ν 1/ = t ν 1 ( 1 1 t ) ν 1/ = m= Γ(1/ ν + m) m! Γ(1/ ν) tν 1 m 5

w(z) = z ν = m= C e izt (t 1) ν 1/ dt z ν Γ(1/ ν + m) m! Γ(1/ ν) C e izt t ν 1 m dt t = ±1 t = t = C 1.5 D D arg z i e i arg z Re(z) > (t 1) ν+1/ e izt D zt = ue 3πi/ e izt t ν 1 m dt = e izt t ν 1 m dt C D = e izt t ν 1 m dt D = z m ν ( 1) m+1 e νπi L e u ( u) ν 1 m du L 1.5 (1.6) (1.7) e u ( u) ν 1 m du = i sin π(ν m) Γ(ν m) L = πi Γ(m + 1 ν) (1.4) w(z) = πi e νπi m= = πi e νπi m= = πi π e νπi Γ(1/ ν) ν ( z J ν (z) = Γ(1/ ν) πi π ( 1) m z m ν Γ(1/ ν + m) m! Γ(1/ ν)γ(m + 1 ν) ( 1) m z m ν m! Γ(1/ ν) e νπi ( z ) ν ) ν C m= π m ν Γ(m ν + 1) ( 1) m m! Γ(m ν + 1) ( z ) m e izt (t 1) ν 1/ dt, Re(z) > ν + 1/ 6

1.1 Re(z) > (1.55) (1.56) 1.6 Γ(1/ ν) ( z ν { J ν (z) = πi π ) } + C 1 C Γ(1/ ν) ( z ) ν { } J ν (z) = πi e νπi π C 1 D 1 S C D Riemann S 1 S Riemann S n 1.9 t D 1 + S (t 1) ν 1/ = S 1 (t 1) ν 1/ e πi(ν 1/) e izt (t 1) ν 1/ dt = e πi(ν 1/) D D C e izt (t 1) ν 1/ dt = e πνi C e izt (t 1) ν 1/ dt J ν (z) = Γ(1/ ν) πi π ( z ν {e ) νπi C 1 +e νπi C } (1.38), (1.38) H ν (1) (z) = J ν(z) e νπi J ν (z) i sin νπ H ν () (z) = eνπi J ν (z) J ν (z) i sin νπ J ν J ν 1 1 1 1 C C 1 D D 1 1.6: 7

H ν (1) (z) = H () ν (z) = Γ(1/ ν) πi π Γ(1/ ν) πi π ( z ) ν ( z ) ν C 1 e izt (t 1) ν 1/ dt C e izt (t 1) ν 1/ dt (1.57) Re(z) >, Re(ν) + 1/ > ν + 1/ H (1) ν (z) S H () ν (z) S 1 1.13 f(z), φ(z) f < A φ, z > δ A, δ ɛ f = O(φ) f < ɛ φ, z > δ δ f = o(φ) z f/φ f O(φ) f/φ f o(φ) g(s) = e sf(z) φ(z) dz L s f(z), φ(z) z s f(z) f(z) s f(z) f(z) = u(x, y) + i v(x, y), z = x + iy f(z) u(x, y) u x + u y = f (z ) = z = z u/ x =, u/ y = u(x, y) 8

u(x, y) s f(z) s 1.13 f(z), φ(z) z s g(s) g(s) = e sf(z) φ(z) dz (1.58) L s [ φ(z ) e sf(z ) π ] 1/ (1.59) s f (z ) f(z) f(z) = f(z ) + a (z z ) + a 3 (z z ) 3 +, z z = R f(z) M a n M R n f(z) = f(z ) + a (z z ) + F (z) { F (z) z z 3 M + z z 4 } + = M z z 3 R R 3 R 4 R 3 {R z z }, z z < R ɛ > z z s ɛ R/ s F (z) M R 3 s 3ɛ e sf(z) = exp{s f(z ) + s a (z z ) } {1 + O(s 1 3ɛ )} ɛ > 1/3 s φ(z) φ(z) = φ(z ) + Φ(z) 9

Φ(z) M R 1 s ɛ 1/3 < ɛ < 1/ φ(z) = φ(z ) + O(s ɛ ) = φ(z ) + o(s 1 3ɛ ) z z < R { } φ(z ) e sf(z ) e sa (z z ) 1 + O(s 1 3ɛ ) dz (1.6) L a = A e iα, A > ; z z = r e iθ, s ɛ < r < s ɛ a (z z ) = A r e i(α+θ) α + θ = π θ s ɛ φ(z ) e sf(z )+i(π α)/ e Asr s ɛ dr t = A s r, T = (A s 1 ɛ ) 1/ = O(s 1/ ɛ ) s ɛ s ɛ e Asr dr = (As) 1/ T T e t dt = (As) 1/ { π T } e t dt e t dt T T ] [ e t e t dt = t T e T 1 1 + T T e t t dt < e T T O(s 3ɛ 3/ ) (1.6) φ(z ) e sf(z )+i(π α)/ (As) 1/{ π + O(s 3ɛ 3/ ) }{ 1 + O(s 1 3ɛ ) } z = z (1.58) s [ φ(z ) e sf(z ) π ] 1/ s f (z ) 3

arg(z) < π Γ(z) = e t t z 1 dt t t z exp{ i arg(z)} Γ(z) = z z e z(log t t) t 1 dt t = 1 (1.59) Γ(z) π z z 1/ e z, z, arg(z) < π (1.61) (Stirling) 1.14 (Darboux) 1.1 ( (Darboux) ) f(z) a a + h f(a + h) f(a) = n 1 m=1 h m m! f (m) (a) + hn (n 1)! 1 f (n) (a + th) (1 t) n 1 dt (1.6) n 1 h m m! (1 t)m f (m) (a + th) t m=1 d n 1 h m dt m! (1 t)m f (m) (a + th) m=1 n 1 { h m = (m 1)! (1 t)m 1 f (m) (a + th) + hm+1 m! m=1 ( m 1 = p ) n = = p= h p+1 p! (1 t) p f (p+1) (a + th) + n 1 m=1 h m+1 m! h n (n 1)! (1 t)m 1 f (n) (a + th) h f (1) (a + th) } (1 t) m f (m+1) (a + th) (1 t) m f (m+1) (a + th) 31

t 1 f(a + h) f(a) = n 1 m=1 h m m! f (m) (a) + hn (n 1)! 1 f (n) (a + th) (1 t) n 1 dt 1 C 1 (+) 1.7: 1. π < arg(z) < π, Re(ν) + 1/ > ν + 1/ H ν (1) (z) = { n 1 Γ(ν + m + 1/) πz ei(z π/4 νπ/) m! Γ(ν m + 1/) m= ( i ) m } + Rn z (1.63) R n sin(ν + 1/)π = ( i ) n (+) z i (n 1)! Γ(ν n + 1) 1 ( e u ( u) ν 1/ u n (1 t) n 1 1 + iut ) ν n 1/ dt du z (+) 1.7 (1.57) H ν (1) (z) = Γ(1/ ν) πi π (z ) (1+) ν 1+ i e izt (t 1) ν 1/ dt 3

t 1 = i u z 1 H ν (1) (z) = = Γ(1/ ν) (+) π πz i Γ(1/ ν) (+) π e i(z π/4 νπ/) πz e iz e u ( i) ν 1/ ( u) ( ν 1/ 1 + iu ) ν 1/ du z e u ( u) ν 1/ ( 1 + iu z ) ν 1/ du (1.64) f(z) = (1 + z) ν 1/, a =, h = iu/z ( iu ) ν 1/ 1 + f() z n 1 1 = m! m=1 ( iu ) m f (m) () + z f (m) () = ( ν 1 ( iu 1 + z n 1 = + m= ) ν 1/ 1 m! 1 (n 1)! 1 (n 1)! ( iu z ) n 1 f (n) ( iut z ) (1 t)n 1 dt )( 3) ( 1) Γ(ν + 1/) ν ν m + = Γ(ν m + 1/) ( iu ) m Γ(ν + 1/) z Γ(ν m + 1/) ( iu ) 1 n z Γ(ν + 1/) Γ(ν n + 1/) H ν (1) (z) = + i Γ(1/ ν) (+) e i(z π/4 νπ/) π πz { n 1 e u ( u) ν 1/ m= Γ(ν + 1/) (n 1)! Γ(ν n + 1/) ( 1 + iut z ) ν n 1/(1 t) n 1 dt Γ(ν + 1/) ( iu ) m m! Γ(ν m + 1/) z ( iu ) 1 ( n 1 + iut ) ν n 1/ (1 t) dt} n 1 du z z (1.6) (+) e u ( u) ν 1/ u m du = ( 1) m ( i) sin π(ν + m + 1/) H ν (1) (z) e u u ν+m 1/ du = i ( 1) m Γ(ν + m + 1/) sin π(ν + m + 1/) 33

Γ(1/ ν)γ(ν + 1/) = π e i(z π/4 νπ/) πz { n 1 Γ(ν + m + 1/) ( i ) m m! Γ(ν m + 1/) sin π(ν + m + 1/) z m= i ( i ) n (+) + e u ( u) ν 1/ u n (n 1)! Γ(ν n + 1) z 1 ( (1 t) n 1 1 + iut ) ν n 1/ } dt du z Γ(1/ ν)γ(ν + 1/) = π/ sin π(ν + 1/) R n sin(ν + 1/)π i = (n 1)! Γ(ν n + 1) ( i ) n (+) 1 ( e u ( u) ν 1/ u n (1 t) n 1 1 + iut ) ν n 1/ dt du z z H ν (1) (z) = { n 1 Γ(ν + m + 1/) πz ei(z π/4 νπ/) m! Γ(ν m + 1/) m= ( i ) m } + Rn z (1.57) Re(z) > H (1) ν Re(izt) < z = re iθ, t = ρe iα Re(izt) = r ρ sin(θ + α) < < θ+α < π. α 1 π < α < π π < θ < π (1.64) (1.63) π < arg(z) < π H ν () (z) = { n 1 Γ(ν + m + 1/) πz e i(z π/4 νπ/) m! Γ(ν m + 1/) m= ( i ) m } + R n z (1.65) (1.65) π < arg(z) < π 34

1.14 π < arg(z) < π, Re(ν) > 1/ J ν (z) = Y ν (z) = πz {cos(z π 4 νπ [n/] ) sin(z π 4 νπ [(n 1)/] ) πz k= k= {sin(z π 4 νπ [n/] ) + cos(z π 4 νπ [(n 1)/] ) k= ( 1) k Γ(ν + k + 1/) (k)! Γ(ν k + 1/) ( 1) k Γ(ν + k + 3/) (k + 1)! Γ(ν k 1/) k= ( 1) k Γ(ν + k + 1/) (k)! Γ(ν k + 1/) ( 1) k Γ(ν + k + 3/) (k + 1)! Γ(ν k 1/) lim R n (k) z n 1 = (k = 1, ) z ( 1 ) k z ( 1 ) k+1 + Rn (1)} z ( 1 ) k z (1.66) ( 1 ) k+1 + Rn ()} z (1.67) Re(ν) 1/ (1.17) (1.36) J ν (z) = 1 ν 1/ l J ν (z) = Y ν (z) = πz { } H (1) ν (z) + H () ν (z), Y ν (z) = i { } H (1) ν (z) H () ν (z) (l =, 1,,... ) (1.63), (1.65) {cos(z π 4 νπ [n/] ) sin(z π 4 νπ [(n 1)/] ) πz k= k= {sin(z π 4 νπ [n/] ) + cos(z π 4 νπ [(n 1)/] ) k= ( 1) k Γ(ν + k + 1/) (k)! Γ(ν k + 1/) ( 1) k Γ(ν + k + 3/) (k + 1)! Γ(ν k 1/) k= ( 1) k Γ(ν + k + 1/) (k)! Γ(ν k + 1/) ( 1) k Γ(ν + k + 3/) (k + 1)! Γ(ν k 1/) (1.63) R n lim R n (k) z n = (k = 1, ) z ( 1 ) k z ( 1 ) k+1 + Rn (1)} z ( 1 ) k z ( 1 ) k+1 + Rn ()} z ν 1/ = l (l =, 1,,... ) J ν (z) (1.3), (1.4) (1.17) 35

Y ν (z) (1.9) (1.) n l R n (z) = 36

.1 3 P (x, y, z), Q(x, y, z ) V (x, y, z) V (x, y, z) = ( ) 1 { P Q = (x x ) + (y y ) + (z z ) } 1/ V x + V y (.1) + V z = (.) (Laplace) (r, θ, ϕ) x = r sin θ cos ϕ y = r sin θ sin ϕ (.3) z = r cos θ r θ ϕ Q (,, r ) P Q = x + y + (z r ) = r rr cos θ + r V V = { r rr cos θ + r } { 1/ = r 1 1 r cos θ + ( r ) } 1/ r r (.4) r > r V (r/r ) { 1 r 1 r cos θ + ( r ) } 1/ 1 = r r r n= P n (cos θ) ( r r ) n r < r (.5) 37

.1 (.5) P n (cos θ) ζ = cos θ ζ n n (Legendre) ζ = cos θ, r/r = α (.5) ( ) 1 αζ + α 1/ = α n P n (ζ) α < 1 (.6) n= ζ = 1 (1 α + α ) 1/ = (1 α) 1 = ζ = 1 α = P n (1) = 1 n =, 1,,... P n ( 1) = ( 1) n P (ζ) = 1 (.6) α ζ α 1 αζ + α = n= nαn 1 P n (ζ) n= αn P n (ζ) n= n= (ζ α) α n P n (ζ) = (1 αζ + α ) nα n 1 P n (ζ) α n n = 1, (n + 1)P n+1 (ζ) np n 1 (ζ) = (n + 1)ζP n (ζ) (.7) n= P 1 (ζ) = ζ, P (ζ) = (3ζ 1)/ P 1, P, P 3,... P n (ζ) ζ (.7) (.6) ζ d [ ] P n+1 (ζ) P n 1 (ζ) = (n + 1)P n (ζ) (.8) dζ α n 38

1 P P 1 P P 3 P 4 P 5 x.1: P n (ζ) (.6) P n (cos θ). P n (cos θ) k 1 (s)!(4k s)! ((k)!) P k (cos θ) = cos[(k s)θ] + 4k 1 (s!) ((k s)!) 4k (k!) 4 s= k (s)!(4k + s)! P k+1 (cos θ) = cos[(k + 1 s)θ] 4k+1 (s!) ((k + 1 s)!) s= (.9) (1 α cos θ + α ) = (1 αe iθ )(1 αe iθ ) z < 1 1 1 z = (.6) r= Γ(r + 1/) r!γ(1/) zr α n P n (cos θ) = n= r= s= Γ(r + 1/)Γ(s + 1/) α r+s e i(r s)θ r!s!γ(1/)γ(1/) 39

α n P n (cos θ) = n r= Γ(r + 1/)Γ(n r + 1/) e i(r n)θ πr!(n r)! (1.4) z = n r + 1/ z = r + 1/ e i(r n)θ Γ(n r + 1/) = (n r)! π (n r) (n r)! Γ(r + 1/) = (r)! π r (r)! Γ(r + 1/)Γ(n r + 1/) πr!(n r)! = (r)!(n r)! n (r!) ((n r)!) P n (cos θ) = n r= (r)!(n r)! n (r!) ((n r)!) ei(r n)θ (.1) r = s r = n s (s)!(n s)! cos[(n s)θ] n 1 (s!) ((n s)!) n k + 1 k k P k (cos θ) = P k+1 (cos θ) = k 1 s= (s)!(4k s)! ((k)!) cos[(k s)θ] + 4k 1 (s!) ((k s)!) 4k (k!) 4 k s= (s)!(4k + s)! cos[(k + 1 s)θ] 4k+1 (s!) ((k + 1 s)!) (.1) P n (cos θ) = θ = P n (1) = n c r e i(r n)θ r= n r= c r 4

P n (1) = 1 n c r = 1 r= P n (cos θ) n c r = 1 (.11) r=. (.) (r, θ, ϕ) V r + r V r + 1 ( V r θ V (.5) V + cot θ θ + 1 V ) sin = (.1) θ ϕ V = 1 r n= P n (cos θ) ( r r ) n r < r V r = 1 r V r = 1 r 3 n= n= np n (cos θ) ( r r ) n 1 n(n 1)P n (cos θ) ( r r ) n r < r P n (cos θ) 1 (.1) ( r ) n { n(n + 1)P n + d P n r dθ n= + cot θ dp } n = dθ r P n d P n dθ + cot θ dp n dθ + n(n + 1)P n = (.13) θ ζ = cos θ d { (1 ζ ) dp } n + n(n + 1)P n = (.14) dζ dζ 41

P n (ζ) ζ.3 P n (ζ) P n (ζ) = σ s= ( 1) s (n s)! n s!(n s)!(n s)! ζn s (.15) σ n n/ (n 1)/ P n (ζ) = (.14) m k(k 1)a k ζ k + k= m a k ζ k k= m a k ζ k{ n(n + 1) k(k + 1) } = k= ζ ζ m n(n + 1) = m(m + 1) (.16) m = n, n 1 m = n 1 P n (ζ) ζ m = n ζ k ( k < n) a k+ (k + )(k + 1) = a k (k n)(k + n + 1) a k = (k + )(k + 1) (k n)(k + n + 1) a k+ (.17) a n s = ( 1)s (n!) (n s)! (n s)!s!(n s)!(n)! a n a n = (n)!/ n (n!) a n s = ( 1)s (n s)! n s!(n s)!(n s)! σ ( 1) s (n s)! P n (ζ) = n s!(n s)!(n s)! ζn s s= σ n n/ (n 1)/ 4

(Rodrigues).4 ( (Rodrigues) ) P n (ζ) = 1 n n! d n [ ] (ζ 1) n dζ n (.18) (Rodrigues) d n dζ n [ζn s ] = (n s)! (n s)! (.15) σ ( 1) s P n (ζ) = n s!(n s)! s= n (ζ 1) n = nc s ζ n s ( 1) s = s= n 1 d n σ n! dζ n [(ζ 1) n ] = P n (ζ) = 1 n n! s= d n ζ n s dζ n [ζn s ] n s= ( 1) s (n s)!s! d n [ ] (ζ 1) n dζ n ( 1) s n! (n s)!s! ζn s d n dζ n [ζn s ].3 f(ζ) 1 ζ 1 I = (.18) 1 1 1 f(ζ)p n (ζ) dζ I = 1 f(ζ) dn n n! 1 dζ n [(ζ 1) n ] dζ = 1 [ d n 1 ] 1 n n! dζ n 1 [(ζ 1) n ]f(ζ) 1 1 f (ζ) dn 1 1 n n! 1 dζ n 1 [(ζ 1) n ] dζ 43

f(ζ) ζ = ±1 n 1 1 f(ζ)p n (ζ) dζ = ( 1)n n n! 1 1 (ζ 1) n f (n) (ζ) dζ (.19) f(ζ) = ζ m m : f (n) (ζ) = { Γ(m + 1)ζ m n /Γ(m n + 1) m n m < n m n 1 1 P n (ζ)ζ m dζ = Γ(m + 1) n n!γ(m n + 1) 1 1 (1 ζ ) n ζ m n dζ m n m n m n 1 1 (1 ζ ) n ζ m n dζ = B ( n + 1, m n + 1 ) Γ(n + 1)Γ(m/ n/ + 1/) = Γ(m/ + n/ + 3/) m < n 1 P n (ζ)ζ m m! Γ(m/ n/ + 1/) dζ = m n : ( ) 1 n (m n)! Γ(m/ + n/ + 3/) m n : ( ) (.19) f(ζ) = P m (ζ).5 P n (ζ) 1 1 1 1 P n (ζ)p m (ζ) dζ = m n (.) P n (ζ) dζ = n + 1 (.1) P m (ζ) m d n P m (ζ) dζ n = n > m 44

1 1 P n (ζ)p m (ζ) dζ = n > m m, n 1 1 P n (ζ)p m (ζ) dζ = m n P n (ζ) P m (ζ) m = n (.18) d n P n (ζ) dζ n (.19) 1 1 P n (ζ) dζ = (n)! ( n n!) 1 1 = (n)! n n! = (n)! Γ(1/)Γ(n + 1) ( n n!) Γ(n + 3/) = n + 1 (1 ζ ) n dζ = (n)! B(n + 1, 1/) ( n n!) = (n)! (n!) n+1 ( n n!) (n + 1)!.4 ν d { (1 z ) df } + ν(ν + 1)f = (.) dz dz.6 P ν (z) = Γ(ν + m + 1) ( z 1 ) m, z 1 < (.3) (m!) Γ(ν + 1 m) m= z 1 < ν z = 1 t (.) t(1 t) d f + (1 t)df + ν(ν + 1)f = dt dt 45

f(t) = t α m= a m t m t a α = a m+1 (m + α + 1) + a m { ν(ν + 1) (m + α)(m + α + 1) } = α = (ν m)(ν + m + 1) a m+1 = a (m + 1) m a 1, a,... a a = 1 ν(ν + 1)(ν 1)(ν + ) f(t) = 1 ν(ν + 1) t + t 1 ν(ν + 1)(ν + )(ν + 3)(ν 1)(ν ) t 3 + 1 3. a m a m = ( 1)m ν(ν + 1) (ν + m)(ν 1) (ν m + 1) (.4) (m!) = ( 1)m Γ(ν + m + 1) Γ(ν 1) (m!) Γ(ν 1) Γ(ν m + 1) = ( 1)m Γ(ν + m + 1) (m!) Γ(ν m + 1) t = 1 z Γ(ν + m + 1) ( z 1 P ν (z) = (m!) Γ(ν + 1 m) m= lim a m (m + 1) = lim = 1 m a m+1 m (ν + m + 1)(m ν ) P ν (z) z 1 < z = 1 (.3) ν u(x) u(1) = f() = 1 P n (1) = 1 f P n a m (.4) a m = 1 Γ(m + ν + 1) (m!) Γ(ν + 1) Γ(m ν) Γ( ν) ) m 46

P ν (z) = P ν 1 (z) (.5).7 ν P ν (z) z = 1 z = 1 (.) z = t 1.6 ν(ν + 1)(ν 1)(ν + ) f(t) = 1 ν(ν + 1) t + t 1 ν(ν + 1)(ν + )(ν + 3)(ν 1)(ν ) t 3 + 1 3 α 1, α f 1 (t) f(t) f 1 (t) = f(t) log t + g(t), g(t) = b m t m m=1 ( z < 1 ) (p1) [ ] f 1 ( z + 1 ) z 1 z + 1 < (.3) z 1 < { } z : z 1 <, z + 1 < Pν (z) f f 1 P ν (z) = A f( z + 1 ) + B f 1( z + 1 ) A, B (.3) z + 1 < P ν (z) = A P ν ( z) + B { P ν ( z) log 1 + z + g( z + 1 } ), z + 1 < (.6) ν B z = 1 f(1) = m= Γ(ν + m + 1) (m!) Γ(ν m + 1) = m= Γ(m + ν + 1)Γ(m ν) (m!) Γ(ν + 1)Γ( ν) (p339) [ ] < M ν Γ(ν + m + 1)Γ(m ν), m 1 m (m!) M ν f(1) P ν (z) z = 1 B (.6) ν z = 1 47

ν n P n (z) B = (.15) P n (z) = ( 1) n P n ( z).5 z C f(z) n f (n) (z) z C f (n) (z) = n! f(t) dt n =, 1,,... (.7) πi (t z) n+1 f(z) = (z 1) n n n! P n (z) = 1 n πi C C (t 1) n dt n =, 1,,... (.8) (t z) n+1 (Schläfli) (.7) n ν P ν (z) F (z) = 1 ν πi C (t 1) ν dt (.9) (t z) ν+1 df dz = (ν + 1) ν πi C (t 1) ν (t z) dt, d F (ν + 1)(ν + ) = ν+ dz ν πi C (t 1) ν dt (t z) ν+3 (1 z ) d F df z + ν(ν + 1)F dz dz (ν + 1) (t 1) ν = ν πi (t z) ν+3 (νt νtz + ν + tz) dt = C (ν + 1) ν πi C d [ (t 1) ν+1 ] dt dt (t z) ν+ (t 1) ν+1 C (t z) ν+ (.3) (t 1) ν+1 ν t (t z) ν+ (t 1) ν+1 ν t (t z) ν+ 48

z C 1 1 S.: t (.9) Riemann ν Riemann t t = z, 1, t arg t 1 = nπ t z S n. z 1 1 S n S n+1 S z 1 1 C Riemann F (z). C P ν (z) = 1 (t 1) ν dt (, 1] ν πi (t z) (.3) ν+1 C P ν (z) (.3) P ν (z) (.3).8 z 1 < 1 ν πi C (t 1) ν (t z) dt = ν+1 m= Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( z 1 ) m (.31) C.. C R < t = 1 t 1 = Re iθ, θ < π z 1 < R ( (t z) ν 1 = (t 1) ν 1 1 z 1 t 1 ) ν 1 49

= (t 1) ν 1 {1 + m=1 Γ(ν + m + 1) m! Γ(ν + 1) ( z 1 ) m } t 1 (t 1) ν (t z) ν+1 = (t + 1)ν (t 1) ν (t z) ν 1 = (t + 1) ν (t 1) 1 = (t + 1) ν m 1 Γ(ν + m + 1) (t 1) (z 1)m m!γ(ν + 1) m= ( (t + 1) ν = ν 1 + t 1 ) ν = ν r= C (t + 1) ν (t 1) m 1 dt = ν 1 ν πi C (t 1) ν (t z) dt = ν+1 m= r= ( z 1 ) m Γ(ν + m + 1) t 1 m! Γ(ν + 1) m= Γ(ν + 1) ( t 1 r r! Γ(ν + 1 r) ) r Γ(ν + 1) (t 1) r m 1 dt r r! Γ(ν + 1 r) C = πi ν m Γ(ν + 1) m! Γ(ν + 1 m) Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( z 1 ) m, z 1 < R <.6 (.16) m = n 1 (.17) a n 3 = (n + 1)(n + ) (n + 3) a n 1, a n 5 = (n + 3)(n + 4) (n + 5)4 a n 3,... a n (k+1) = (n + 1)(n + ) (n + k) (n + 3)(n + 5) (n + k + 1) k k! a n 1 n 1 n 5

n ν π Γ(ν + 1) a n (k+1) z n (k+1) a ν 1 = ν+1 Γ(ν + 3/) k= Q ν (z) = π Γ(ν + 1)z ν 1 ν+1 Γ(ν + 3/) { 1 + k=1 (ν + 1)(ν + ) (ν + k) } k k! (ν + 3)(ν + 5) (ν + k + 1) z k (.3) z > 1 (.3) T (z) = 1 (t 1) ν dt (.33) ν+ i sin νπ D (z t) ν+1 D P ν (z) D C.3 D (t 1) ν (z t) ν t = 1 t = 1 t = z (.34) P arg(t 1) = z D 1 D.3: 1 P arg(z t) = arg(z) t +1 z D.3 1 (t 1) ν Q ν (z) = dt (, 1] ν+ i sin νπ (z t) (.34) ν+1 D Q ν (z) P ν (z) (.34) Q ν (z) z > 1 (.3) 51

.9 z > 1 1 ν+ i sin νπ D = (t 1) ν dt (z t) ν+1 1 { ν+1 z ν+1 k= Γ(ν + k + 1)Γ(k + 1/) (k)! Γ(ν + k + 3/) D.3 z k } (.35) Re(ν + 1/) >.3 t = t = 1 t = 1 t = 1 t = t = t = 1 t = 1 t = 1 t = 1 e iνπ (1 t ) ν (z t) ν+1 1 e iνπ (1 t ) ν (z t) ν+1 1 e iνπ (1 t ) ν (z t) ν+1 1 e iνπ (1 t ) ν 1 ν+ i sin νπ (z t) ν+1 Re(ν + 1/) > ±1 { 1 Q ν (z) = (e iνπ e iνπ ) 1 + ν+ i sin νπ (eiνπ e iνπ ) 1 1 = {e iνπ ν+ i sin νπ 1 1 (1 t ) ν dt (z t) ν+1 (1 t ) ν } (z t) dt ν+1 (1 t ) ν dt eiνπ (z t) ν+1 1 1 = ν 1 (1 t ) ν (z t) ν 1 dt Re(ν + 1/) > 1 z > 1 (z t) ν 1 = z {1 ν 1 Γ(ν + k + 1) ( t ) k } + k! Γ(ν + 1) z k=1 1 (1 t ) ν } (z t) dt ν+1 1 t 1 1 { 1 Q ν (z) = (1 t ) ν Γ(ν + k + 1) 1 } dt + t k (1 t ) ν dt ν+1 z ν+1 k! Γ(ν + 1)z k 1 k=1 1 5

k (, 1) 1 1 Q ν (z) = 1 { ν+1 z ν+1 t k (1 t ) ν dt = k= Γ(k + 1/)Γ(ν + 1) Γ(ν + k + 3/) Γ(ν + k + 1)Γ(k + 1/) (k)! Γ(ν + k + 3/) z k } z > 1 (.3) ν z > 1 ν ν.7 z < 1 (.3).4.1 (.3) P ν (z) = e νπi (z+,1+,z,1 ) (t 1) ν dt (.36) 4π ν sin νπ (t z) ν+1.4 c z b a d 1 1.4: (z+,1+,z-,1-) d a πν c b πν 53

1 < Re(ν) < t = z, t = 1 + + + = (1 e πνi ) +(1 e πνi ) a b c d a b = (1 e πνi ) a+b = e πνi (t 1) ν i sin νπ dt (t z) ν+1 P ν (z) = 1 ν πi a+b a+b (t 1) ν dt (t z) ν+1 e νπi (t 1) ν P ν (z) = dt 4π ν sin νπ a+b+c+d (t z) ν+1 P ν (z) ν 1 < Re(ν) <.4 z 1 z 1 (z+, 1+, z, 1 ) P ν (z) = e νπi 4π ν sin νπ (z+,1+,z,1 ) (t 1) ν dt (t z) ν+1.4 (t 1) ν.5 (t z) ν+1.11 z + 1 < (z+, 1+,z, 1 ) = e ±νπi ν m= (t 1) ν dt (t z) ν+1 ( 1) m Γ(ν + 1) m!γ(ν + 1 m) ( z + 1 ) m (1+,+,1, ) u ν (u 1) ν 1 u m du (.37).5 A ( 1, 1) t 1 A π A t = 1, t = 1, t = z L, M, N (z+, 1+, z, 1 ), (z+, 1+, z, 1 ) L, M, N 54

z 1 A 1.5: (z+,1+,z,1 ) (t 1) ν (t z) dt = N + ν+1 Me π(ν+1)i Ne πνi M (z+, 1+,z, 1 ) (t 1) ν (t z) dt = N + ν+1 Le π(ν+1)i Ne πνi L (.38) ( 1+,1 ) (t 1) ν dt (t z), (.34) Q ν+1 ν(z) Im(z) > (L M)e πνi Im(z) < L M ( 1+,1 ) (t 1) ν dt (t z) ν+1 = e πνi 1 e πνi ( (z+,1+,z,1 ) (z+, 1+,z, 1 )) (t 1) ν dt, Im(z) > (.39) (t z) ν+1 ( 1+,1 ) = (t 1) ν dt (t z) ν+1 1 ( (z+,1+,z,1 ) 1 e πνi (z+, 1+,z, 1 )) (t 1) ν dt, Im(z) < (.4) (t z) ν+1 (.36) P ν (z) (z+, 1+,z, 1 ) (t 1) ν dt t + 1 = (z + 1)u (t z) ν+1.6 1 1 (1+, +, 1, ) ν (1+,+,1, ) u ν ( z + 1 Im(z) z + 1 u 1) ν(u 1) ν 1 du u 1 = e ±πi ( 1 z + 1 ) u 55

1 (z + 1)u/ u = (1 (z + 1)u/) ν (z + 1)/ < 1 (z+, 1+,z, 1 ) = e ±νπi ν m= (t 1) ν dt (t z) ν+1 ( 1) m Γ(ν + 1) m!γ(ν + 1 m) ( z + 1 ) m (1+,+,1, ) u ν (u 1) ν 1 u m du W T V U 1.6: (1+,+,1-,-).1 (1+,+,1, ) u ν (u 1) ν 1 u m du = ( 1) sin πνi Γ(ν + m + 1)Γ( ν) νπ e m! (.41).6 u 1 = xe iθ 1 < Re(ν) < u = 1 u = θ = π π(ν + 1) ) u ν+m (u 1) ν 1 du = (e 1 π(ν+1)i e π(ν+1)i (1 x) ν+m x ν 1 dx + = i sin νπ Γ(ν + m + 1)Γ( ν) m! π(ν + m) π(ν + m) u ν+m (u 1) ν 1 du = i e πνi Γ(ν + m + 1)Γ( ν) sin νπ m! + u ν+m (u 1) ν 1 du = ( 1) sin πνi Γ(ν + m + 1)Γ( ν) νπ e m! + + + 56

z < 1.13 z 1 <, z + 1 < Q ν (z) = π {e νπi sin νπ m= m= ( 1) m Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( 1) m Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( 1 z ( 1 + z ) m } Im(z) >, Im(z) < + (.4) ) m (z+, 1+,z, 1 ) (t 1) ν (t z) ν 1 dt (1+,+,1, ) ( z + 1 ) ν(u = ν u ν u 1 1) ν 1 du = e ±νπi ν ( 1) m Γ(ν + 1) ( z + 1 ) m (1+,+,1, ) u ν (u 1) ν 1 u m du m!γ(ν + 1 m) m= = ν+ πe νπi sin νπ m= ( 1) m Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( z + 1 ) m (Im(z) < e νπi 1 ) ν ν ν (.34) Q ν (z) z 1 <, z + 1 < 1 (t 1) ν Q ν (z) = dt ν+ i sin νπ D (z t) ν+1 e νπi ( 1+,1 ) = (t 1) ν (t z) ν 1 dt ν+ i sin νπ e 3νπi { = e νπi 4π ν sin νπp ν+ i sin νπ(1 e νπi ν (z) ) ν+ πe νπi ( 1) m Γ(ν + m + 1) ( z + 1 ) m } sin νπ (m!) Γ(ν + 1 m) m= (Im(z) < e νπi 1 ) 57

= π {e νπi sin νπ m= ( 1) m Γ(ν + m + 1) ( 1 z ) m (m!) Γ(ν + 1 m) ( 1) m Γ(ν + m + 1) ( 1 + z ) m } (m!) Γ(ν + 1 m) m= Im(z) >, Im(z) < + ν ν 1 P ν 1 (z), Q ν 1 (z) P ν (z), Q ν (z), P ν 1 (z), Q ν 1 (z).14 (, 1] P ν (z) = 1 π tan νπ{ Q ν (z) Q ν 1 (z) } (.43) (.3) ν ν 1 z 1 < P ν 1 (z) = P ν (z) (.44) P ν (z) (.3) z 1 < P ν (z) z (.44) P ν (z) z (.4) ν ν 1 P ν (z) ν ν 1 Q ν 1 (z) = π {e (ν+1)πi sin νπ m= m= ( 1) m Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( 1) m Γ(ν + m + 1) (m!) Γ(ν + 1 m) ( 1 z ( 1 + z ) m } (.4) π cos νπp ν (z) = sin νπ { Q ν (z) Q ν 1 (z) } ) m 58

(, 1] ν Q ν (z) = Q ν 1 (z) ν P ν (z) = 1 π tan νπ{ Q ν (z) Q ν 1 (z) } Q ν (z) z > 1 (.3), z > 1 P ν (z) P ν (z) = { 1 + + tan νπ Γ(ν + 1) π(z) ν+1 Γ(ν + 3/) (ν + 1)(ν + ) (ν + k) 1 ) k } k k!(ν + 3) (ν + k + 1)( z k=1 Γ(ν + { 1/)(z)ν Γ(ν + 1) 1 + π k=1 ( ν)( ν + 1) ( ν + k 1 ) k } k k!( ν + 1) ( ν + k 1)( z z > 1 (.45).8.15 P ν (z)q ν (z) P ν (z)q ν (z) = 1 1 z (.46) d { (1 z ) df } + ν(ν + 1)f = dz dz P ν (z), Q ν (z) d { } (1 z )P ν Q ν d { } (1 z )Q ν P ν = dz dz d { } P ν (z)q dz ν (z) P ν (z)q ν (z) = z { } P 1 z ν (z)q ν (z) P ν (z)q ν (z) 59

P ν (z)q ν (z) P ν (z)q ν (z) = c 1 z c : z z c (.45) P ν (z) (.34) Q ν (z) Γ(ν + 1/) Γ(ν + 1) π (z)ν z 1 π Γ(ν + 1) Γ(ν + 3/) (z) ν+1 z 1 (ν + 1/)Γ(ν + 1/) P ν (z)q ν (z) P ν (z)q ν (z) Γ(ν + 3/) c 1 P ν (z)q ν (z) P ν (z)q ν (z) = 1 1 z z.16 (ν + 1)Q ν+1 (z) + νq ν 1 (z) = (ν + 1)zQ ν (z) (.47) (.35) z > 1 Q ν (z) ν ν +1 (ν +1) (ν + 1)Q ν+1 (z) = (ν + 1) ν z ν = (ν + 1) ν z ν k=1 k= Γ(ν + k + )Γ(k + 1/) (k)! Γ(ν + k + 5/) Γ(ν + k)γ(k 1/) 4(k )! Γ(ν + k + 3/) z k ν ν 1 ν νq ν 1 (z) = ν ν z nu k= Γ(ν + k)γ(k + 1/) (k)! Γ(ν + k + 1/) z k z k 6

{ Γ(ν)Γ(1/) ν (ν + 1)Q ν+1 (z) + νq ν 1 (z) = ν z ν Γ(ν + 1/) νγ(ν + k)γ(k + 1/) (ν + 1)Γ(ν + k)γ(k 1/) + (k)! Γ(ν + k + 1/) z k + 4(k )! Γ(ν + k + 3/) k=1 k=1 z k } νγ(ν + k)γ(k + 1/) (ν + 1)Γ(ν + k)γ(k 1/) + (k)! Γ(ν + k + 1/) 4(k )! Γ(ν + k + 3/) Γ(ν + k)γ(k 1/) (ν + k)(ν + 1/) = Γ(ν + k + 1/)(k )! 4k(ν + k + 1/) (ν + 1/)Γ(ν + k + 1)Γ(k + 1/) = (k)! Γ(ν + k + 3/) (ν + 1)Q ν+1 (z) + νq ν 1 (z) = (ν + 1)zQ ν (z) z > 1 (.34) z < 1.17 Q ν+1 (z) zq ν = (ν + 1)Q ν (z) (.48) zq ν (z) Q ν 1 (z) = νq ν (z) (.49).17 ν 1 (ν + 1)Γ(ν + 1)Γ(1/) Q ν+1 (z) zq ν (z) = z ν 1 Γ(ν + 3/) { (k + ν + 1)Γ(k + ν + 1)Γ(k + 1/) + ν 1 (k)! Γ(ν + k + 3/) k=1 (ν + k)γ(ν + k)γ(k 1/) (k )! Γ(ν + k + 3/) ν 1 (ν + 1)Γ(ν + 1)Γ(1/) = Γ(ν + 3/) + ν 1 k=1 } z k ν 1 z ν 1 (ν + 1)Γ(ν + k + 1)Γ(k + 1/) (k)! Γ(ν + k + 3/) z k ν 1 61

= (ν + 1) { ν 1 z ν 1 k= Γ(ν + k + 1)Γ(k + 1/) (k)! Γ(ν + k + 3/) z k } Q ν+1 (z) zq ν = (ν + 1)Q ν (z) (.47) (.48) zq ν (z) Q ν 1 (z) = νq ν (z) P ν (z).18 (ν + 1)P ν+1 (z) + νp ν 1 (z) = (ν + 1)zP ν (z) (.5) (.43) (ν + 1)P ν+1 (z) + νp ν 1 (z) = ν + 1 tan νπ { Q ν+1 (z) Q ν (z) } + ν π π tan νπ{ Q ν 1 (z) Q ν (z) } (ν + 1) (ν + 1) = tan(νπ) zq ν (z) tan(νπ) zq ν 1 (z) π π (ν + 1)P ν+1 (z) + νp ν 1 (z) = (ν + 1)zP ν (z) {zpν (z) P ν 1 (z) = νp ν (z) P ν+1 (z) zp ν (z) = (ν + 1)P ν (z) (.51) (.3) ν = { Q (z) = z 1 1 + 1 3z + 1 5z + 1 } 4 7z + z > 1 6 1 log z + 1 z 1 = 1 z + 1 3z + 1 3 5z + z > 1 5 6

Q (z) = 1 log z + 1 z 1 ν = 1 z > 1 Q 1 (z) = 1 3z + 1 5z 4 + 1 7z 6 +, z > 1 Q 1 (z) = z log z + 1 z 1 1 (.47) z > 1 Q (z) = 1 P (z) log z + 1 z 1 3 z z > 1 1 Q Q Q 1 1 x.7:.9 m 1 < x < 1 P ν m (x) = (1 x ) m/ dm P ν (x) dx m, Q ν m (x) = (1 x ) m/ dm Q ν (x) dx m (.5) ν v = dm f dx m (1 x ) d f df x + ν(ν + 1)f = dx dx 63

m (1 x ) d v (m + 1)xdv + (ν m)(ν + m + 1)v = dx dx w = (1 x ) m/ v (1 x ) d w { } dx xdw dx + ν(ν + 1) m w = (.53) 1 x P m ν (x), Q m ν (x).19 P m n (x), P m r (x) 1 1 P n m (x)p r m (x)dx = r n (.54) P m n (x), P m r (x) d [ { (1 x ) P m r (x) dp n m (x) dx dx P m n (x) dp r m (x) }] dx + { n(n + 1) r(r + 1) } P m n (x)p m r (x) = x [ { (1 x ) P m r (x) dp n m (x) dx P m n (x) dp r m (x) }] x=1 dx x= 1 = { r(r + 1) n(n + 1) } 1 1 P n m (x)p r m (x)dx P m n (x), P m r (x) x = ±1 r n 1 1 P n m (x)p r m (x)dx = r n. (ν + 1)xP ν m (x) = (ν m + 1)P ν+1 m (x) + (ν + m)p ν 1 m (x) (.55) 64

(.51) P ν+1 (z) P ν 1 (z) = (ν + 1)P ν (z) { d (ν + 1)xP m ν (x) = (ν + 1)(1 x ) m/ m dx (xp ν) m dm 1 P } ν m dx { m 1 d = (1 x ) m/ m [ ] d m 1 [ ] } (ν + 1)Pν+1 + νp dx m ν 1 m Pν+1 P dx m 1 ν 1 { = (1 x ) m/ (ν m + 1) dm P ν+1 dx + (ν + P } ν 1 m m)dm dx m (ν + 1)xP ν m (x) = (ν m + 1)P ν+1 m (x) + (ν + m)p ν 1 m (x) P m ν 1 (x) = xp m ν (x) (ν m + 1) 1 x P m 1 ν (x) P m ν+1 (x) = xp m ν (x) + (ν + m) 1 x P m 1 ν (x) 1 x P m+1 ν (x) = mxp m ν (x) (ν + m)(ν m + 1) 1 x P m 1 ν (x) (.56).1 P m n (x) 1 1 [ m Pn (x) ] dx = n + 1 (n + m)! (n m)! ( m n) (.57) (.55) ν = n + 1 P m n (x) (n + m + 1) = (n + 3) = 1 1 1 1 (n + 3)(n m + 1) (n + 1) [ m Pn (x) ] 1 dx = (n + 3) xp m n+1 (x)p m n (x) dx [ P m 1 n+1 (x) (n + 1) 1 [ dx P m n+1 (x)] 1 1 { (n m + 1)P n+1 m (x) + (n + m)p n 1 m (x) 1 [ ] dx (n + 3) P m (n + m + 1) 1 [ n+1 (x) = (n + 1) m Pn (x) ] dx 1 (n m + 1) 1 }] dx 65

(n + m + 1) (n + m) = (n 1) (n m + 1) (n m) = 1 (n + m + 1)(n + m) (m + 1) (n m + 1)! 1 [ Pn 1 m (x) ] dx (m + 1) 1 1 [ Pm m (x) ] dx d m P m (x) = (m)! dx m m m! 1 1 [ Pm m (x) ] dx = [ (m)! ] = m (m!) [ ] (m)! (n + 3) 1 1 1 1 m (m!) m (1 x ) m dx [ Γ(m + 1) ] Γ(m + ) = (m)! m + 1 [ m Pn+1 (x) ] (n + m + 1)! dx = (n m + 1)! 1 [ m Pn (x) ] dx = n + 1 1 (n + m)! (n m)!.1 P m n (x) P n (x) = 1 (t 1) n dt n πi (t x) n+1 C P m n (x) = (1 x m/ (n + m)! (t 1) n ) dt n πi n! C (t x) n+m+1 C x (1 x ) x C x 1 1 C C t x = 1 x e iφ φ < π P n m (x) = (n + m)! π n! π (x + 1 x i sin φ) n e imφ dφ 66

φ, π π/, 5π/ P n m (x) = (n + m)! e mπi/ π n! π (x + 1 x i cos φ) n e imφ dφ (.58) (Laplace) 1 x 1 x = cos θ x θ 1 1 P n m (cos θ) = (n + m)! e mπi/ π n! π (cos θ + i sin θ cos φ) n e imφ dφ (.59) (.58) t = e iφ D π (cos θ + i sin θ cos φ) n e imφ dφ = 1 [cos θ + i ( i D sin θ t + 1 )] nt m 1 dt t = 1 e nφ(t) dt i D n φ(t) = log [{cos θ + i ( sin θ t + 1 )} nt (m+1)] t n dφ dt = i sin θ (1 1 t ) cos θ + i sin θ (1 + 1 m + 1 t ) n m n t = ±1 (1.59) 1 [ π ] 1/ 1 [ π ] 1/ i enφ(1) + nφ (1) i enφ( 1) nφ ( 1) = 1 [ π(cos θ + i sin θ) ] 1/ i (cos θ + i sin θ)n ni sin θ + 1 [ π(cos θ i sin θ) ] 1/e i (cos θ i sin θ)n πi(m+1) ni sin θ ( π ) 1/ [ = cos (n + 1 mπ )θ n sin θ π ] e mπi/ 4 m n (m + n)! n! n m ( P m n (cos θ) n m ) 1/ [ cos (n + 1 mπ )θ nπ sin θ π ], n (.6) 4 1 t.11 ν P ν (cos θ) = (.3) P 1/+iσ (cos θ) = 1 + 1/4 + σ (1!) sin θ + (1/4 + σ )(9/4 + σ 4 ) (!) sin 4 θ + (.61) 67

P 1/+iσ (cos θ) (σ : ) σ = ix (x : ) P 1/ x (cos θ) = 1 + 1/4 x (1!) sin θ + (1/4 x )(9/4 x 4 ) (!) sin 4 θ + x 1/4 P 1/ x (cos θ) P s (cos θ) = 1 s. θ π θ ν P ν (cos θ) = (.14) d { ( (1 ζ dp ν ) P dζ ν dζ P dp )} { } ν ν = ν (ν + 1) ν(ν + 1) P dζ ν P ν, ζ = cos θ ζ = 1 1 (ν ν)(ν + ν + 1) P ν (ζ)p ν (ζ) dζ cos θ { } = sin θ P ν (cos θ)p ν (cos θ) P ν (cos θ)p ν (cos θ) (.6) P ν (cos θ) = ν (.3) P ν (cos θ) = P ν (cos θ) ν ν P ν (cos θ) = (.6) ν + ν + 1 = 1 cos θ P ν (ζ)p ν (ζ) dζ = ν ν P ν (ζ)p ν (ζ) 1 cos θ P ν (ζ)p ν (ζ) dζ > ν + ν + 1 = ν ν Re(ν) = 1/ (.61) σ P 1/+iσ (cos θ) ν P ν (cos θ) = ν P ν m (cos θ) = m : (.6) n( ) P ν m (cos θ) Γ(ν + m + 1) Γ(ν + 1) ( νπ sin θ ) 1/ [ (ν 1) mπ cos + θ π 4 ] 68

arg(ν) π ɛ, ɛ > (.63) ν (, ) P m ν (cos θ) = 69

3 3.1 L p (Ω) p 1 p < R n Ω u ( 1/p u L p u(x) dx) p < (3.1) Ω L p (Ω) u = v (a.e.) u v L p (Ω) 3.1 ( (H ölder) ) 1 < p < q 1 = 1 p 1 u(x)v(x) dx u L p v L q u L p (Ω), v L q (Ω) (3.) Ω u L p = u L q = u L p u L q a, b = ab p 1 a p + q 1 b q (3.3) a = u(x) / u L p, b = v(x) / v L q u(x) v(x) u L p u L q 1 p u(x) p u p L p + 1 q v(x) q v q L q 7

Ω u(x) v(x) u L p u L q dx 1 p 1 u p L p Ω u(x) p dx + 1 q 1 u q L q Ω v(x) q dx = 1 3. ( (Minkowski) ) 1 p < ( u(x) + v(x) p ) 1/p ( dx Ω u, v L p (Ω) u + v L p (Ω) u(x) p dx Ω ) 1/p + ( v(x) p 1/p dx) (3.4) Ω p = 1 p > 1 (a + b) p p 1 (a p + b p ) (a, b ) u(x) + v(x) p ( ) p ) u(x) + v(x) p 1( u(x) p + v(x) p u + v L p (Ω) u(x) + v(x) p dx u(x) + v(x) p 1 u(x) dx + Ω Ω u(x) + v(x) p 1 v(x) dx p 1 + q 1 = 1 q = p/(p 1) u + v p 1 L q (Ω) u(x) + v(x) p 1 ( u(x) dx u(x) + v(x) (p 1)q ) 1/q ( dx u(x) p ) 1/p dx Ω Ω Ω u(x) + v(x) p 1 ( v(x) dx u(x) + v(x) (p 1)q ) 1/q ( dx v(x) p ) 1/p dx Ω Ω Ω u(x) + v(x) p ( dx u(x) + v(x) p dx Ω Ω [( u(x) p ) 1/p ( dx + Ω Ω ) 1/q v(x) p dx Ω ) 1/p ] 71

u(x) + v(x) p ( dx u(x) + v(x) p 1/q dx) Ω Ω u(x) + v(x) p dx = Ω 3.1 L p (Ω) u L p L p (Ω) L p (Ω) 3. ( ) p ) u(x) + v(x) p u(x) + v(x) p 1( u(x) p + v(x) p u + v L p (Ω) u L p {u n } {n nk } u nk+1 u nk L p< 1 k (k = 1,,... ) n, m > n 1 u n u m L p< 1 n 1 n, m > n u n u m L p< 1 n (> n 1 ) n 1, n,... {u nk } v k = u nk 1 v k+1 v k L p< = 1 (3.5) k k=1 k=1 g n g 1 (x) = v 1 (x) x Ω n 1 g n (x) = v 1 (x) + v k+1 (x) v k (x) n, x Ω k=1 g L p (Ω) (3.6) 7

g n (x) p, g n (x) p g n+1 (x) p, g n (x) v n (x) {g n (x) p } Ω ( n 1 ) p ( g n (x) p dx = g n p L p v 1 L p + v k+1 v k L p k=1 ) p v 1 L p +1 g n (x) Ω g(x) = lim n g n (x) a.e. x Ω Ω ( g(x) p dx g L p (Ω) ) p v 1 L p +1 {v k } k < m v m (x) v k (x) m 1 v j+1 (x) v j (x) = g m (x) g k (x) (3.7) j=k {g m (x)} x Ω lim v m (x) v k (x) = m,k lim n v n(x) = u (x) u a.e. x Ω u L p (Ω) (3.7) k = 1 g 1 (x) m v m (x) v 1 (x) + g m (x) u (x) v 1 (x) + g(x) v 1, g L p (Ω) u L p (Ω) v k (x) u (x) p ( v k (x) + u (x) ) p 73

( ) p g k (x) + u (x) ( ) p } g(x) + u (x) p 1{(g k (x)) p + u (x) p vk (x) u (x) p Ω k p 1 { (g k (x)) p + u (x) p } v k (x) u (x) p (k ) v k u L p (k ) {u n } ɛ > N n, n k > N = u n u nk = u n v k < ɛ k u n u < ɛ u n u L p (n ) 3. C X x, y (x, y) 1) 3) (x, y) 1) (x, x) (x, x) = x = ) (x, y) = (y, x) 3) (x 1 + x, y) = (x 1, y) + (x, y) (α x, y) = α (x, y) α C X X (x, y 1 + y ) = (x, y 1 ) + (x, y ) 74

(x, α y) = α (x, y) α j, β j C x j, y j X (j = 1,,..., n) ( n n ) α j x j, β k y k = j=1 k=1 n n α j β k (x j, y k ) j=1 k=1 3. X x x = (x, x) 1/ X 1) x ) x = x = 3) αx = α x (α C) 4) x + y x + y 1) 3) 4) (x, y) x y (x, y) = (x, y) α = (x, y) / y (x + α y, x + α y) = x +α (y, x) + α (x, y) + α y = x (x, y) / y (x, y) x y x + y = (x + y, x + y) = x +(x, y) + (y, x)+ y x + x y + y = ( x + y ) 75

4) (x n, y n ) (x, y) = (x n x, y n y) + (x n x, y) + (x, y n y) (x n x, y n y) x n x y n y x n x, y n y x n x, y n y (x n x, y n y) (x n x, y), (x, y n y) (x n, y n ) (x, X X Ω R n u, v L (Ω) (u, v) = u(t) v(t) dt (3.8) L (Ω) Ω 3.3 X A, B x A, y B (x, y) = A B X L L L L = {x X; x L} L X α, β C, x, y L (αx + βy, L) = α (x, L) + β (y, L) = αx + βy L 76

L X u n L, u n = u X (u, L) = lim n (u n, L) = u L L 3.3 ( ) X L X x X x = y + z y L z L y x L P L x = y x = y + z = y + z y, y L z, z L y y = z z L L = {} y = y, z = z x X X L δ δ = inf ξ L x ξ ξ n L, x ξ n δ {ξ n } {ξ n } L (x ξ n )+(x ξ m ) + (x ξ n ) (x ξ m ) = x ξ n + x ξ m 1 (ξ n + ξ m ) L δ x ξ n + ξ m ξ n ξ m = (x ξ n ) (x ξ m ) = x ξ n + x ξ m (x ξ n ) + (x ξ m ) = x ξ n + x ξ m 4 x ξ n + ξ m 77

x ξ n + x ξ m 4δ m, n x ξ n δ, x ξ m δ ξ n ξ m {ξ n } {ξ n } y X L y L δ = x y z = x y z L ξ L ϕ(t) = z γ t ξ γ = (z, ξ), t R y + γ t ξ L δ = ϕ() ϕ(t) ϕ(t) = z γ t (ξ, z) γ t (z, ξ)+ γ t ξ = z γ t + γ t ξ γ t ϕ(t) < ϕ() = δ γ = z L X L X = L L X {x k } X (x j, x k ) = δ jk (j, k = 1,,..., n) {x k } ( n n ) n α j x j, β k x k = α k β k j=1 k=1 1) L ( (, 1) ) { sin πjt} j=1 ) L ( (, 1) ) {e πkti } k= (i = 1) k=1 3.4 ( ) {x k } X x X (x, x k ) x k 78

α k = (x, x k ) n x α k x k k=1 = x = x n α k (x, x k ) k=1 n α k k=1 n α k (x k, x) + k=1 n α j α k (x j, x k ) n α k x n k=1 j,k=1 {x k } X {x k } L { n } L = α k x k : α k C, n k=1 L = L L X X L X = L L x X x = y + z (y L, z L ) y = P L x 3.5 {x k } L L y = α k x k (α k C) k=1 n L α j x j, j=1 n α j x j j=1 m α j x j, j=1 m ( m α j x j = j=1 (m > n) j=n+1 α j x j, m k=n+1 α k x k ) = m j=n+1 α j 79

m n n, m α j α j x j L L j=n+1 y = α j x j L j=1 j=1 3.6 {x k } L x, x X P L x = k (P L x, P L x ) = k (x, x k )x k (3.9) (x, x k )(x, x k ) (3.1) α j = (x, x j ) y = α j x j L j=1 (x y, x k ) = (x, x k ) ( α j x j, x k ) = (x, x k ) j=1 α j (x j, x k ) = n x y x k {x k } γ k x k (γ k C) (x y, ξ) = ξ L z x y L P L x = y = α j x j = (x, x j )x j j=1 j=1 P L x = n (x, x k )x k, k k k=1 k x, x X α j = (x, x j ), α j = (x, x j ) ( n n ) α j x j, α jx n ( n ) 1/ ( n ) 1/ j α j α j α j α j x x j=1 j=1 j=1 n α j α j j=1 ( n (P L x, P L x ) = lim α j x j, n j=1 j=1 n ) α jx j = j=1 j=1 j=1 k=1 α j α j = (x, x j ) (x, x j ) j=1 j=1 8

(P L x, P L x ) = (x, x k ) (x, x k ) k 3.4 3.7 {x k } {x k } L (1) (5) (1) L = X () x X x = (x, x k ) x k k (3) x X x = k (x, x k ) ( ) (4) x, x X (x, x ) = k (x, x k ) (x, x k ) (5) x X (x, x k ) = x = (1) () L = X x = P L x = k (x, x k )x k () (4) () (x, x ) = (P L x, P L x ) = k (x, x k ) (x, x k ) (4) (3) x = (x, x) = (P L x, P L x) = k (x, x k ) (x, x k ) = k (x, x k ) (3) (5) (5) (1) x L (x, x k ) = (k = 1,... ) x = L = {} X = L X {x k } (1) (5) {x k } X 3.8 L ( ( π, π) ) { 1 π, 1 π sin nt, 1 π cos nt } n=1 ϕ (t) = 1 π, ϕ k (t) = 1 π cos nt, ϕ k 1 (t) = 1 π sin nt 81

ϕ k (t) (k =, 1,,... ) x L ( π, π) π π x(t) ϕ k (t) dt = (k =, 1,... ) = x(t) x(t) x(t) x(t) x(t) x(t) x(t) [ π, π] x(t) t t x(t ) x(t ) > x(t) ε >, δ > I = [t δ, t + δ] x(t) ε (t I) h(t) = 1 + cos(t t ) cos δ h(t) 1 (t I), h(t) < 1 (t [ π, π] \ I) h n (t) ϕ j π π x(t) h n (t) dt = (n = 1,,... ) (3.11) [ π, π] I [ π, π] \ I I h(t) < 1 (t I ) x(t) h n (t) dt x(t) dt (3.1) I I I = [t δ, t + δ ] ( < δ < δ) h(t) 1 + η (t I ) η > x(t) h n (t) ε (1 + η) n (t I ) x(t) h n (t) dt I x(t) h n (t) dt δ ε (1 + η) n I n (1 + η) n (3.11) x(t) h n (t) dt = I x(t) h n (t) dt I (3.1) x(t) h n (t) dt x(t) I x(t) y(t) = t π x(s) ds 8

y(t) [ π, π] x ϕ y(π) = y( π) = π y(t) cos kt dt = 1 [ ] π y(t) sin kt 1 π x(t) sin kt dt = π k π k π π π y(t) sin kt dt = cos kt, sin kt z(t) = y(t) π π y(s)ds {ϕ k } z(t) z(t) z (t) = y (t) = x(t) x(t) (a.e.) L ( ( π, π) ) x(t) x(t) = 1 ( ) α + α n cos nt + β n sin nt n=1 (3.13) α k, β k π π α k = 1 x(t) cos kt dt, β k = 1 π π π π x(t) sin kt dt α k, β k 3.5 X {y i } {x k } { a) xn y 1,..., y n (3.14) b) y n x 1,..., x n n x 1 = y 1 y 1 x = y (y, x 1 )x 1 y (y, x 1 )x 1 83

n 1 y n (y n, x j )x j j=1 x n = n 1 y n (y n, x j )x j j=1 {y j } y 1 x 1 x 1 y 1 y y 1 x 1 y (y, x 1 )x 1 x y 1, y x 1, x,..., x n 1 y 1, y,..., y n 1 n 1 y n (y n, x j )x j j=1 x n x n x 1, x,..., x n 1, y n x n y 1, y,..., y n b) {x k } {x k } 1 e iθ k 3.9 X X X {z j } z 1 = z 1 z z 1 z n z 1, z,..., z n 1 {z j } {y j } {y j } a), b) {x k } x X (x, x k ) = (k = 1,,... ) y k x 1, x,... x k (x, y k ) = (k = 1,,... ) z k y 1, y,... y k (x, z k ) = (k = 1,,... ) 84

{z k } X z kn x (n ) {z kn } (x, x) = lim n (x, z kn ) = x = {x k } 1 t 1 1, t, t,..., t n,... L ( 1, 1) y n (t) = t n 1 y 1 = x 1 (t) = 1 (y, x 1 ) =, y = 3 x (t) = y (y, x 1 )x 1 3 y (y, x 1 )x 1 = t {x k } P n (t) = 1 n n! d n dt n (t 1) n P n (t) n 1 1 P n (t)p m (t) dt = n + 1 δ nm {P n } x n+1 (t) = ( n + 1 ) 1/Pn (t) (3.15) {x n } (3.14) (3.14) (3.15) {x n } 3.6 J ν (αx), J ν (βx) d [ x dj ν(αx) dx dx J ν (βx) x dj ν(βx) dx ] J ν (αx) = (β α ) x J ν (αx)j ν (βx) (3.16) 85

ν > 1 1 1 J ν (αx)j ν (βx) x dx = 1 { } α J β α ν (α)j ν (β) β J ν (α)j ν (β) (3.17) α = β β = α + ɛ ɛ 1 [J ν (αx)] x dx = 1 = 1 { [J ν (α)] J ν (α)j ν (α) 1 } α J ν(α)j ν (α) { ) } [J ν (α)] + (1 ν [J α ν (α)] J ν (x) = (> ) α, β 1 J ν (αx)j ν (βx) x dx = 1 α β [J ν (αx)] x dx = 1 [J ν (α)] = 1 [J ν+1(α)] (3.18) J ν (x) = j 1, j,... J ν (j m ) = < j 1 < j < j 3 < {j m } x J ν+1 (j m ) J ν(j m x) (m = 1,,... ) L ( (, 1) ). G.N.Watson [18] x =, 1. 3 3.1 T n (t, x) = n m=1 J ν (j m x)j ν (j m t) J ν+1 (j m ) ( < x < 1, ν + 1/ ) (3.19) x lim n t ν+1 T n (t, x) dt = 1 xν, 1 lim t ν+1 T n (t, x) dt = 1 n xν (3.) x. 86

G.N.Watson [18] [18.1],[18.] 3. (, 1) xf(x) (a, b) (a, b) [, 1] x / [a, b], < x < 1 n b a tf(t)t n (t, x) dt = o(1) (3.1) T n (t, x) 3.1 G.N.Watson [18] The analogue of Riemamm-Lebesgue Lemma 3.3 n U t t ν+1 T n (t, x) dt < U < x 1, < t 1 (3.) T n (t, x) 3.1 G.N.Watson [18] [18.] 3.1 (, 1) xf(x) f(x) (a, b) < a < b < 1 (a, b) x 1 { } f(x + ) + f(x ) = a m = J ν+1(j m ) m=1 1 a m J ν (j m x) t f(t) J ν (j m t) dt (ν + 1/ ) (3.3) 3.1 n a m J ν (j m x) = m=1 1 1 { } f(x + ) + f(x ) t f(t) T n (t, x) dt (3.4) 87

= lim n x ν f(x ) x t ν+1 T n (t, x) dt + lim n x ν f(x + ). x } S n (x) t {t ν+1 ν f(t) x ν f(x ) T n (t, x) dt + 1 x 1 x t ν+1 T n (t, x) dt (3.5) } t {t ν+1 ν f(t) x ν f(x + ) T n (t, x) dt (3.6) (3.3) n S n (x). 1 x } t {t ν+1 ν f(t) x ν f(x + ) T n (t, x) dt. t ν f(t) x ν f(x + ) (x, b) (x, b) t χ 1 (x + ) = χ (x + ) = χ 1 (t), χ (t) t ν f(t) x ν f(x + ) χ 1 (t) χ (t). Modern Analysis [3.64] ε > < δ < b x δ x t x + δ χ 1 (t) < ε, χ (t) < ε. 1 } t {t ν+1 ν f(t) x ν f(x + ) T n (t, x) dt x = + 1 x+δ x+δ x } t {t ν+1 ν f(t) x ν f(x + ) T n (t, x) dt t ν+1 χ 1 (t) T n (t, x) dt x+δ x t ν+1 χ (t) T n (t, x) dt. 3. n 1 } t {t ν+1 ν f(t) x ν f(x + ) T n (t, x) dt < ε (3.7) x+δ. x+δ x t ν+1 χ 1 (t) T n (t, x) dt = χ 1 (x + δ) x+δ x+ξ t ν+1 T n (t, x) dt ξ (, δ). 3.3 n U χ 1 (x + δ) x+δ x+ξ t ν+1 T n (t, x) dt < ε U 88