Improper Integrals of Real Functions
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Note: It is important to note the following: -
If the improper integral converges, then the Cauchy principal value exists. Further, . -
When the Cauchy principal value exists, it is not always true that converges. For example, exists, but fails to exist. -
If is an even function on (That is, for all ) and if the Cauchy principal value exists, then the improper integral exists and equal to .
Note: If is not an even function, then it is always a good practice to write explicitly before the integral symbol to denote the Cauchy principal value of the improper integral. |
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