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Principal part

From Wikipedia, the free encyclopedia

In mathematics, the principal part has several independent meanings but usually refers to the negative-power portion of the Laurent series of a function.

Laurent series definition

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The principal part at of a function

is the portion of the Laurent series consisting of terms with negative degree.[1] That is,

is the principal part of at . If the Laurent series has an inner radius of convergence of , then has an essential singularity at if and only if the principal part is an infinite sum. If the inner radius of convergence is not , then may be regular at despite the Laurent series having an infinite principal part.

Other definitions

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Calculus

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Consider the difference between the function differential and the actual increment:

The differential dy is sometimes called the principal (linear) part of the function increment Δy.

Distribution theory

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The term principal part is also used for certain kinds of distributions having a singular support at a single point.

See also

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References

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  1. ^ Laurent. 16 October 2016. ISBN 9781467210782. Retrieved 31 March 2016.
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