Ward identities for 1D integrals
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Some integrals of type
$$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$
can be solved by realizing that they satisfy a differential equation
like
$$ left(a - d^2/da^2right)I(a)=0,$$
and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
but the same can be done for more general integrands of the similar form.
My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
i.e. something like
$$ I(a_1, ldots, a_n)
= int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$
Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?
integration differential-equations
asked Nov 19 at 15:04
z.v.
189111
add a comment |
up vote
2
down vote
favorite
Some integrals of type
$$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$
can be solved by realizing that they satisfy a differential equation
like
$$ left(a - d^2/da^2right)I(a)=0,$$
and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
but the same can be done for more general integrands of the similar form.
My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
i.e. something like
$$ I(a_1, ldots, a_n)
= int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$
Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?
integration differential-equations
asked Nov 19 at 15:04
z.v.
189111
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Some integrals of type
$$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$
can be solved by realizing that they satisfy a differential equation
like
$$ left(a - d^2/da^2right)I(a)=0,$$
and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
but the same can be done for more general integrands of the similar form.
My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
i.e. something like
$$ I(a_1, ldots, a_n)
= int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$
Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?
integration differential-equations
asked Nov 19 at 15:04
z.v.
189111
Some integrals of type
$$ I(a)=int_{-infty}^infty dx~ e^{i(x^3/3 + a x)},$$
can be solved by realizing that they satisfy a differential equation
like
$$ left(a - d^2/da^2right)I(a)=0,$$
and then we can solve the differential equation given some initial value. In this particular example, we have the usual Airy equation,
but the same can be done for more general integrands of the similar form.
My question is: is there a general way how to apply these methods for a case when I have a general polynomial in the integrand,
i.e. something like
$$ I(a_1, ldots, a_n)
= int_{-infty}^infty dx~ e^{a_1 x + ldots +a_n x^n} ?$$
Whenever such integral is well defined it seems I can write a set of partial differential equations that determine it and it looks like I can do these things in a non-unique way. Is there a general theory that governs how to do these things systematically and optimally?
integration differential-equations
integration differential-equations
asked Nov 19 at 15:04
z.v.
189111
asked Nov 19 at 15:04
z.v.
189111
asked Nov 19 at 15:04
z.v.
189111
asked Nov 19 at 15:04
z.v.
189111
asked Nov 19 at 15:04
z.v.
189111
189111
add a comment |
add a comment |
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