n | 0 | 1 | 2 | 3 |
x^n (-(2^(-n) cos(n π) Γ(-n))/π + (2^(-n - 2) cos(n π) Γ(-n) x^2)/(π n + π) - (2^(-n - 5) cos(n π) Γ(-n) x^4)/((n^2 + 3 n + 2) π) + O(x^6)) + x^(-n) (-(2^n Γ(n))/π + (2^(n - 2) Γ(n) x^2)/(π - n π) - (2^(n - 5) Γ(n) x^4)/((n^2 - 3 n + 2) π) + O(x^6))
cos(1/4 π (2 n + 1) - x) (((4 n^2 - 1) (1/x)^(3/2))/(4 sqrt(2 π)) + ((5 - 2 n) (2 n - 3) (2 n + 3) (2 n + 5) (4 n^2 - 1) (1/x)^(7/2))/(1536 sqrt(2 π)) + O((1/x)^(11/2))) + sin(1/4 π (2 n + 1) - x) (-sqrt(2/π) sqrt(1/x) - (((3 - 2 n) (2 n - 1) (2 n + 1) (2 n + 3)) (1/x)^(5/2))/(64 sqrt(2 π)) - (((2 n - 7) (2 n - 5) (2 n - 3) (2 n - 1) (2 n + 1) (2 n + 3) (2 n + 5) (2 n + 7)) (1/x)^(9/2))/(49152 sqrt(2 π)) + O((1/x)^(11/2)))
d/dx(Y_n(x)) = 1/2 (Y_(n - 1)(x) - Y_(n + 1)(x))
integral Y_n(x) dx = 2^(-n - 1) (x^(n + 1) cot(π n) Γ((n + 1)/2) _1 F^~_2((n + 1)/2 ;n + 1, (n + 3)/2;-x^2/4) - 4^n x^(1 - n) csc(π n) Γ(1/2 - n/2) _1 F^~_2(1/2 - n/2;1 - n, 3/2 - n/2;-x^2/4)) + constant
lim_(x-> ± ∞) Y_n(x) = 0
Y_n(x) = ((i^n K_n(i x) + (log(i x) - log(x)) J_n(x)) (-1) 2)/π for n element Z
Y_n(x) = csc(π n) (cos(n π) J_n(x) - J_(-n)(x)) for n not element Z
Y_n(x) = ((i^n K_n(i x) + ((log(i x) - log(x)) I_n(i x))/i^n) (-1) 2)/π for n element Z
Y_n(x) = sum_(k=0)^∞ ((x - z_0)^k Y_n^(0, k)(z_0))/(k!) for abs(arg(z_0))<π
Y_n(x) = sum_(k=0)^∞(-((-1)^k 2^(-2 k + n) x^(2 k - n) csc(n π))/(k! Γ(1 + k - n)) + ((-1)^k 2^(-2 k - n) x^(2 k + n) cot(n π))/(k! Γ(1 + k + n))) for n not element Z
Y_n(x) = sum_(k=0)^∞(-((-1)^k 2^(-2 k + n) x^(2 k - n) Γ(n))/(π k! (1 - n)_k) - ((-1)^k 2^(-2 k - n) x^(2 k + n) cos(n π) Γ(-n))/(π k! (1 + n)_k)) for n not element Z
Y_n(x) = -(2^(1 + n) x^(-n))/(sqrt(π) Γ(1/2 - n)) integral_1^∞ (-1 + t^2)^(-1/2 - n) cos(t x) dt for (2 abs(Re(n))<1 and x>0)
Y_n(x) = -( integral_0^∞ e^(-n t - x sinh(t)) (e^(2 n t) + cos(n π)) dt + integral_0^π sin(n t - x sin(t)) dt)/π for 2 arg(x)<π
Y_n(x) = -i/(2 π) integral_(-i ∞ + γ)^(i ∞ + γ) (4^s x^(-2 s) Γ(-n/2 + s) Γ(n/2 + s))/(Γ(1/2 (3 + n - 2 s)) Γ(-1/2 - n/2 + s)) ds for (x>0 and abs(Re(n))/2<γ<3/4)
