Relation to complex analysis

 

Riemann integration[edit]

Main article: Riemann integral

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let ${\displaystyle [�,�]}$ be a closed interval of the real line; then a tagged partition ${\displaystyle \mathcal{�}}$ of ${\displaystyle [�,�]}$ is a finite sequence

${\displaystyle �={�}_0\le {�}_1\le {�}_1\le {�}_2\le {�}_2\le \cdots \le {�}_{�-1}\le {�}_{�}\le {�}_{�}=�.}$

This partitions the interval ${\displaystyle [�,�]}$ into ${\displaystyle �}$ sub-intervals ${\displaystyle [{�}_{�-1},{�}_{�}]}$ indexed by ${\displaystyle �=1,\dots ,�}$, each of which is "tagged" with a distinguished point ${\displaystyle {�}_{�}\in [{�}_{�-1},{�}_{�}]}$. For a function ${\displaystyle �}$ bounded on ${\displaystyle [�,�]}$, we define the Riemann sum of ${\displaystyle �}$ with respect to tagged partition ${\displaystyle \mathcal{�}}$ as

${\displaystyle \sum _{�=1}^{�}�({�}_{�}){\mathrm{\Delta }}_{�},}$

where ${\displaystyle {\mathrm{\Delta }}_{�}={�}_{�}-{�}_{�-1}}$ is the width of sub-interval ${\displaystyle �}$. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, $\Vert {\mathrm{\Delta }}_{�}\Vert =\underset{�=1,\dots ,�}{max}{\mathrm{\Delta }}_{�}$. We say that the Riemann integral of ${\displaystyle �}$ on ${\displaystyle [�,�]}$ is ${\displaystyle �}$ if for any  there exists  such that, for any tagged partition ${\displaystyle \mathcal{�}}$ with mesh , we have

This is sometimes denoted $\mathcal{�}{\int }_{�}^{�}�=�$. When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) Darboux sum. A function is Darboux integrable if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal. In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.

The fundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense.

Lebesgue integration and measure[edit]

Main article: Lebesgue integral

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined. The concept of a measure, an abstraction of length, area, or volume, is central to Lebesgue integral probability theory.

Distributions[edit]

Main article: Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

Relation to complex analysis[edit]

Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line. Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula.

In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers.

Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.