### Differential equations and Laplace transforms: Laplace-transformations

### Riemann-Stieltjes integration

For the purpose of working with delta functions, we introduce a more general approach to definite integration than the *Riemann sums**.*

Riemann-Stieltjes integral Let #f# and #g# be piece-wise continuous functions on a closed interval #\ivcc{a}{b}#. Instead of measuring the area of the region under #f# (as in *Riemann sums*) we consider the following expressions:

\[\sum_{i=0}^{n-1}f(\xi_i)\cdot g\left(x_{i+1}-x_i\right)\] where #\xi_i# is an arbitrary number in the interval #\ivcc{x_i}{x_{i+1}}#. Such a sum is called a **Riemann-Stieltjes sum** of #f# with respect to #g# of granularity #\varepsilon# if #\left|x_{i+1}-x_i\right|\lt\varepsilon# for #i=0,\ldots,n-1#.

If the limit of all Riemann-Stieltjes sums of #f# with respect to #g# of granularity #\varepsilon# exists for #\varepsilon\to0#, then we call it the** Riemann-Stieltjes integral** of #f# with respect to #g# and denote it by

\[\int_a^b f(x)\,\dd g(x)\]

The following result shows that Riemann-Stieltjes integrals generalize Riemann integrals.

Riemann integrals are Riemann-Stieltjes integralsIf #g# is differentiable on #\ivoo{a}{b}#, then \[\int_a^b f(x)\,\dd g(x) = \int_a^b f(x)\cdot g'(x)\,\dd x\]

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