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Graph of
G
i
(
x
)
{\displaystyle \mathrm {Gi} (x)}
H
i
(
x
)
{\displaystyle \mathrm {Hi} (x)}
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
{\displaystyle y''(x)-x\ y(x)={\frac {1}{\pi }}}
and are given by
G
i
(
x
)
=
1
π
∫
0
∞
sin
(
t
3
3
+
x
t
)
d
t
,
{\displaystyle \mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt,}
H
i
(
x
)
=
1
π
∫
0
∞
exp
(
−
t
3
3
+
x
t
)
d
t
.
{\displaystyle \mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt.}
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
.
{\displaystyle {\begin{aligned}\mathrm {Gi} (x)&{}=\mathrm {Bi} (x)\int _{x}^{\infty }\mathrm {Ai} (t)\,dt+\mathrm {Ai} (x)\int _{0}^{x}\mathrm {Bi} (t)\,dt,\\\mathrm {Hi} (x)&{}=\mathrm {Bi} (x)\int _{-\infty }^{x}\mathrm {Ai} (t)\,dt-\mathrm {Ai} (x)\int _{-\infty }^{x}\mathrm {Bi} (t)\,dt.\end{aligned}}}
References Olver, F. W. J. (2010), "Scorer functions" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Scorer, R. S. (1950), "Numerical evaluation of integrals of the form
I
=
∫
x
1
x
2
f
(
x
)
e
i
ϕ
(
x
)
d
x
{\displaystyle I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx}
G
i
(
z
)
=
1
π
∫
0
∞
s
i
n
(
u
z
+
1
3
u
3
)
d
u
{\displaystyle {\rm {Gi}}(z)={\frac {1}{\pi }}\int _{0}^{\infty }{\rm {sin}}\left(uz+{\frac {1}{3}}u^{3}\right)du}
The Quarterly Journal of Mechanics and Applied Mathematics , 3 : 107–112, doi :10.1093/qjmam/3.1.107 , ISSN 0033-5614 , MR 0037604
