Input

F(x)

Plots

Plots

Plots

Alternate form

1/2 sqrt(π) e^(-x^2) erfi(x)

Numerical roots

x = 0

x ≈ 6.29377218667141×10^24...

Series expansion at x=0

x - (2 x^3)/3 + (4 x^5)/15 + O(x^6)

(Taylor series)

Series expansion at x=∞

(1/(2 x) + 1/(4 x^3) + 3/(8 x^5) + O((1/x)^6)) - 1/2 i sqrt(π) e^(-x^2)

Derivative

d/dx(F(x)) = 1 - 2 x F(x)

Indefinite integral

integral F(x) dx = 1/2 x^2 _2 F_2(1, 1;3/2, 2;-x^2) + constant

Local maximum

max{F(x)}≈0.541044 at x≈0.924139

Local minimum

min{F(x)}≈-0.541044 at x≈-0.924139

Limit

lim_(x-> ± ∞) F(x) = 0

Alternative representations

F(x) = (erfi(x) sqrt(π))/(2 e^(x^2))

F(x) = -(i erf(i x) sqrt(π))/(2 e^(x^2))

F(x) = (i erf(i x, 0) sqrt(π))/(2 e^(x^2))

Series representations

F(x) = sum_(k=0)^∞ ((-1)^k x^(1 + 2 k))/(3/2)_k

F(x)∝-(e^(-x^2) sqrt(π) sqrt(-x^2))/(2 x) + ( sum_(k=0)^∞ x^(-2 k) (1/2)_k)/(2 x) for abs(x)->∞

F(x) = 1/2 sqrt(π) z_0 sum_(k=0)^∞ (2^k _2 F^~_2(1, 1;1 - k/2, 3/2 - k/2;-z_0^2) (x - z_0)^k z_0^(-k))/(k!)

F(x) = 1/2 sqrt(π) sum_(j=0)^∞ Res_(s=-1/2 - j) (x^(-2 s) Γ(1/2 - s) Γ(1/2 + s))/Γ(1 - s)

Integral representations

F(x) = e^(-x^2) integral_0^x e^(t^2) dt

F(x) = integral_0^∞ e^(-t^2) sin(2 t x) dt for x element R

F(x) = -(P integral_(-∞)^∞ e^(-t^2)/(t - x) dt)/(2 sqrt(π))

F(x) = -e^(-x^2)/(4 π) integral_(-i ∞ + γ)^(i ∞ + γ) ((i x)^(-2 s) Γ(-s) Γ(1/2 + s))/Γ(1 - s) ds for (γ>-1/2 and abs(arg(i x))<π/2)