Mathematics branches {analysis, mathematics} {mathematical analysis} can be about theory of real-variable functions and theory of integrals and integral equations.
purposes
Mathematical analysis studies continuous but non-differentiable functions. It studies continuous-function series whose sum is discontinuous. It studies continuous functions that are not piecewise monotonic. It studies functions with bounded derivatives that are not Riemann integrable. It studies curves that are rectifiable, but not by calculus arc-length definition. It studies non-integrable functions that are limits of integrable-function series. It studies Fourier series relations to represented functions.
point density
Number of interval points and number of subinterval points are the same.
Laplace transform
Integral from t = -infinity to t = +infinity of e^(-x * t) * g(t) * dt, where g(t) = (0.5 * i) * (integral from x = a - infinity to x = a + infinity of (e^(x * t))*(f(x)) * dx), where a is large.
integral existence
If interval points are differentiable, function can integrate over interval. Intervals have variable maximum and minimum values. Function f(x) has maximum and minimum over interval. Maximum-M limit minus minimum m times x-change dx goes to zero as dx goes to zero: (M - m) * dx.
series expansion
Functions can be equivalent to series. Function series expansions are in integral-equation theory {convergence of mean} {Lebesgue square integral} {Riesz-Fischer theorem} {moment problem} {Holder inequalities} {strong convergence} {weak convergence} {singular integral equations}.
Mathematical Sciences>Calculus>Analysis
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Date Modified: 2022.0224