The concept of the sequence of iterated integrals

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Suppose that −∞≤a<b≤∞, and let f:{a,b}→ℝ be an integrable real function, where {a,b} denote any kind of the finite type intervals or {a,b}=(−∞,b) or (−∞,∞).If a=−∞, then the function f supposed to be integrable in the improper sense. Under the above conditions the following sequence of functions is called the sequence of iterated integrals of f:

 

 

 



 




Example 1

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Let {a,b}=[0,1) and f(s)≡1. Then the sequence of iterated integrals of 1 is defined on [0,1), and

 

 

 



 



Example 2

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Let {a,b}=[-1,1] and f(s)≡1. Then the sequence of iterated integrals of 1 is defined on [-1,1], and

 

 

 



 



Example 3

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Let {a,b}=(-∞,0] and f(s)≡e2s. Then the sequence of iterated integrals of e2s is defined on (-∞,0], and