Neha Patil (Editor)

Scorer's function

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Scorer's function

In mathematics, the Scorer's functions are special functions studied by Scorer (1950) and denoted Gi(x) and Hi(x).

Hi(x) and Gi(x) solve the equation

y ″ ( x ) − x   y ( x ) = 1 π

and are given by

G i ( x ) = 1 π ∫ 0 ∞ sin ⁡ ( t 3 3 + x t ) d t , H i ( x ) = 1 π ∫ 0 ∞ exp ⁡ ( − t 3 3 + x t ) d t .

The Scorer's functions can also be defined in terms of Airy functions:

G i ( x ) = B i ( x ) ∫ x ∞ A i ( t ) d t + A i ( x ) ∫ 0 x B i ( t ) d t , H i ( x ) = B i ( x ) ∫ − ∞ x A i ( t ) d t − A i ( x ) ∫ − ∞ x B i ( t ) d t .

References

Scorer's function Wikipedia
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