An integration scheme appropriate for calculating double integral with variable limits is developed based on the Simpson's rule to evaluate the components of Hamilton's equation.
Although it is difficult to express the function in a closed form without integration, it is a one-dimensional integral with finite limits, and its integrand only composes of elementary (exponential and trigonometric) functions, while the function actually is a two-dimensional integral of exponential function with infinite limits according to (A.2) in Appendix.
To improve this problem the infinite integral is converted into integrals with finite limits.
These SEP expressions consist of single integrals with finite limits and integrand composed of elementary functions only, which can be accurately evaluated numerically.
The symbol error probability expressions obtained contain single integrals with finite limits and integrand composed of elementary functions which help us evaluate our analytical expressions numerically.
It is noted that the obtained expressions for the outage probability and channel capacity given in (28), (30), (31), (33), (37), (38), and (45 - 48) are represented in terms of one-dimensional integrals with definite limits.
The closed-form SEP expressions obtained contain Gaussian -functions and a finite (in proportion to ) sum of single integrals with finite limits and an integrand composed of elementary (exponential, trigonometric, and power) functions, thus readily enabling numerical evaluation.
The obtained closed-form SEP expressions contain a finite (in proportion to ) sum of single integrals with finite limits and an integrand composed of elementary (exponential, trigonometric, and/or power) functions, thus readily enabling numerical evaluation.
The obtained closed-form SEP expressions contain a finite sum of single integrals with finite limits and an integrand composed of elementary (exponential, trigonometric, and/or power) functions, which can be easily and accurately evaluated numerically.
Using the above four integral expressions, the average SEP (24) over fading channels can be conveniently evaluated through numerical integration since these formulae are single integrals with finite limits and an integrand composed of elementary (exponential, trigonometric, and/or power) functions.
He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis.
