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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.
This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right. Conversely, the term "quadrature" may also be used for the solution of differential equations: "solving by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.
The integral here is a complex contour integral which is path-independent because is holomorphic on the whole complex plane . In many applications, the function argument is a real number, in which case the function value is also real.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
A similar effect is available for peak-like functions, such as Gaussian, Exponentially modified Gaussian and other functions with derivatives at integration limits that can be neglected. [9] The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points. [10]
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.
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Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [ 1 ] : 13–15 Other integrals can be approximated by versions of the Gaussian integral.