n | 0 | 1 | 2 | 3 |

n = -m, z = 0, m element Z, m>=1

z = -j_(n, m), m element Z, m>=1

z = j_(n, m), m element Z, m>=1

n>0, z = 0

n = -m, z = 0, m element Z, m>=1

z^n (2^(-n)/Γ(n + 1) - (2^(-n - 2) z^2)/((n + 1) Γ(n + 1)) + (2^(-n - 5) z^4)/((n + 1) (n + 2) Γ(n + 1)) + O(z^6))

sin(1/4 π (2 n + 1) - z) (((4 n^2 - 1) (1/z)^(3/2))/(4 sqrt(2 π)) - (((5 - 2 n) (2 n - 3) (2 n + 3) (2 n + 5) (1 - 4 n^2)) (1/z)^(7/2))/(1536 sqrt(2 π)) + O((1/z)^(11/2))) + cos(1/4 π (2 n + 1) - z) (sqrt(2/π) sqrt(1/z) + ((3 - 2 n) (2 n - 1) (2 n + 1) (2 n + 3) (1/z)^(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/z)^(9/2))/(49152 sqrt(2 π)) + O((1/z)^(11/2)))

d/dz(J_n(z)) = 1/2 (J_(n - 1)(z) - J_(n + 1)(z))

integral J_n(z) dz = 2^(-n - 1) z^(n + 1) Γ((n + 1)/2) _1 F^~_2((n + 1)/2 ;n + 1, (n + 3)/2;-z^2/4) + constant

lim_(z-> ± ∞) J_n(z) = 0

J_n(z) = _0 F^~_1(;1 + n;-z^2/4) (z/2)^n

J_n(z) = (I_n(i z) z^n)/(i z)^n

J_n(z) = (I_n(-i z) (-i i z)^n)/(-i z)^n

J_n(z) = sum_(k=0)^∞ ((-1)^k 2^(-2 k - n) z^(2 k + n))/(k! Γ(1 + k + n))

J_n(z) = (-1)^n sum_(k=0)^∞ ((-1)^k 2^(-2 k + n) z^(2 k - n))/(k! Γ(1 + k - n)) for (n element Z and n<0)

J_n(z) = i^(-n + abs(n)) sum_(k=0)^∞ ((-1)^k 2^(-2 k - abs(n)) z^(2 k + abs(n)))/(k! Γ(1 + k + abs(n))) for n element Z

J_n(z) = 1/π integral_0^π cos(n t - z sin(t)) dt for n element Z

J_n(z) = i^(-n)/π integral_0^π e^(i z cos(t)) cos(n t) dt for n element Z

J_n(z) = 1/(2 π) integral_(-π)^π e^(i (n (-π/2 + t) + z cos(t))) dt for n element Z

J_n(z) = _0 F^~_1(;1 + n;-z^2/4) (z/2)^n

J_n(z) = (_0 F_1(;n + 1;-z^2/4) (z/2)^n)/Γ(n + 1) for (n not element Z or n>=0)

J_n(z) = (_1 F_1(1/2 + n;1 + 2 n;2 i z) z^n)/(Γ(1 + n) 2^n e^(i z))

J_n(z) = ((lim_(a->∞) _1 F_1(a;1 + n;-z^2/(4 a))) z^n)/(Γ(1 + n) 2^n)