If the double integral exists, then it is equal to each of the two iterated integrals ( either " " or " " ) and one often computes it by computing either of the iterated integrals.
12.
Here \ scriptstyle T ^ k [ 1 ] is the k-th iterated integral of the constant 1, that is \ scriptstyle x ^ k / k !, and we get the exponential series.
13.
Then Tonelli's theorem allows us to rewrite H�lder's inequality using iterated integrals : If & thinsp; and are-measurable } } real-or complex-valued functions on the Cartesian product, then
14.
More precisely, the Tonelli Hobson test states that if " & fnof; " is a real-valued measurable function on "'R "'2, and either of the two iterated integrals
15.
If the integral is not absolutely convergent, care is needed not to confuse the concepts of " multiple integral " and " iterated integral ", especially since the same notation is often used for either concept.
16.
Notice that, by convention, the double integral has two integral signs, and the triple integral has three; this is a notational convention which is convenient when computing a multiple integral as an iterated integral, as shown later in this article.
17.
Coordinate iterated integrals ( terms of the signature ) form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and form a natural linear " basis " for continuous functions on paths.
18.
In calculus, interchange of the "'order of integration "'is a methodology that transforms iterated integrals ( or multiple integrals through the use of Fubini's theorem ) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed.
