(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by
(3)
(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand (left) and numerical values of
(red curve in right plot) for small
. The first few values, for , 2, ..., are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955,
....
As ,
(4)
where
(5)
(6)
(OEIS A093601; top curve in right plot above).
The leading asymptotic term was derived by Kac (1943).
Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial Are Real?" Bull. Amer. Math. Soc.32, 1-37, 1995.Kac,
M. "On the Average Number of Real Roots of a Random Algebraic Equation."
Bull. Amer. Math. Soc.49, 314-320, 1943.Kac, M. "A
Correction to 'On the Average Number of Real Roots of a Random Algebraic Equation.'
" Bull. Amer. Math. Soc.49, 938, 1943.Kostlan, E.
"On the Distribution of Roots of Random Polynomials." Ch. 38 in From
Topology to Computation: Proceedings of the Smalefest (Ed. M. W. Hirsch,
J. E. Marsden, and M. Shub). New York: Springer-Verlag, pp. 419-431,
1993. https://doi.org/10.1007/978-1-4612-2740-3_38.Sloane,
N. J. A. Sequence A093601 in "The
On-Line Encyclopedia of Integer Sequences."
of real roots of a random
polynomial of degree 
whose coefficients are
independent, each with the standard normal distribution:
![E_n=1/piint_(-infty)^inftysqrt([(partial^2)/(partialxpartialy)ln(1-(xy)^(n+1))/(1-xy)]_(x=y=t))dt](/images/equations/KacFormula/NumberedEquation1.svg)
(left) and numerical values of
(red curve in right plot) for small
. The first few values, for 
, 2, ..., are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955,
....
,
![2/pi{ln2+int_0^infty[sqrt(1/(x^2)-(4e^(-2x))/((1-e^(-2x))^2))-1/(x+1)]dx}](/images/equations/KacFormula/Inline16.svg)
