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If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by = + (), and = ().
Intuitively, a function = (,) is a function of m even (bosonic, commuting) variables and of n odd (fermionic, anti-commuting) variables. More formally, an element f = f ( x , θ ) ∈ Λ m ∣ n {\displaystyle f=f(x,\theta )\in \Lambda ^{m\mid n}} is a function of the argument x {\displaystyle x} that varies in an open set X ⊂ R m ...
This directly results from the fact that the integrand e −t 2 is an even function (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
These 2 latter inequalities follow from the convexity of the exponential function (or from an analysis of the function ). Letting u = x 2 {\displaystyle u=x^{2}} and making use of the basic properties of improper integrals (the convergence of the integrals is obvious), we obtain the inequalities:
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.
Products of an odd number of Grassmann variables anti-commute with each other; such a product is often called an a-number. Products of an even number of Grassmann variables commute (with all Grassman numbers); they are often called c-numbers. By abuse of terminology, an a-number is sometimes called an anticommuting c-number.