# Gaussian approximation to a certain polynomial

Consider the function: where the independent variable x ranges between 0 and X, and the exponents are large: . [We could call it a “polynomial”, though the exponents need not be integers. Specifically it is the product of “monomials” in x and Xx, so might possibly be called a “sparse” polynomial in this sense.] Surprisingly, it closely resembles a gaussian curve, over our specified domain . Figure: The polynomial with parameters A = 10, B = 13, and X = 11. Our gaussian approximation is visually indistinguishable near the centre. Outside our specified domain the polynomial tends to , and for each non-integer exponent the tail on one side becomes imaginary.

The turning point is where the derivative equals zero. This occurs when x is the surprisingly simple expression: at which the function has value: An arbitrary gaussian, not necessarily normalised, has form: . This has centre D which we equate with , and maximum height C which we set to the above expression. We can fix the final parameter, the standard deviation, by matching the second derivatives at the turning point. Hence the variance is: Hence our gaussian approximation may be expressed: The integral of the original curve turns out to be: This uses the binomial coefficient , which is extended to non-integer values by replacing the factorials with Gamma functions. We could then apply Stirling’s approximation to each factorial, to obtain: though this is more messy to write out. On the other hand, the integral of the gaussian approximation is: We evaluated this integral over all real numbers, because the expression is simpler and still approximately the same. The ratio of the above two expressions is .