How to obtain the explicit solution of the following integral equation

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I'm considering the following integral equation
$$
f(x,y,z)=x+int_0^xint_0^yint_0^z f(u,v,w) dudvdw
$$
It seems that
$$
f(x,y,z)=sum_ngeq 0fracx^n+1 y^n z^n(n+1)!n!n!
$$
is the solution. How to transform it into a closed form instead of a series?







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  • $partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
    – metamorphy
    Aug 12 at 14:28















up vote
2
down vote

favorite
3












I'm considering the following integral equation
$$
f(x,y,z)=x+int_0^xint_0^yint_0^z f(u,v,w) dudvdw
$$
It seems that
$$
f(x,y,z)=sum_ngeq 0fracx^n+1 y^n z^n(n+1)!n!n!
$$
is the solution. How to transform it into a closed form instead of a series?







share|cite|improve this question






















  • $partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
    – metamorphy
    Aug 12 at 14:28













up vote
2
down vote

favorite
3









up vote
2
down vote

favorite
3






3





I'm considering the following integral equation
$$
f(x,y,z)=x+int_0^xint_0^yint_0^z f(u,v,w) dudvdw
$$
It seems that
$$
f(x,y,z)=sum_ngeq 0fracx^n+1 y^n z^n(n+1)!n!n!
$$
is the solution. How to transform it into a closed form instead of a series?







share|cite|improve this question














I'm considering the following integral equation
$$
f(x,y,z)=x+int_0^xint_0^yint_0^z f(u,v,w) dudvdw
$$
It seems that
$$
f(x,y,z)=sum_ngeq 0fracx^n+1 y^n z^n(n+1)!n!n!
$$
is the solution. How to transform it into a closed form instead of a series?









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 12 at 8:02







user529760

















asked Aug 12 at 6:50









Yuhang

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780118











  • $partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
    – metamorphy
    Aug 12 at 14:28

















  • $partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
    – metamorphy
    Aug 12 at 14:28
















$partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
– metamorphy
Aug 12 at 14:28





$partial f / partial x$ is a function of $xyz$, and I wonder what is the "closed form" of it you expect to find (in the two-variable case it would be $I_0(sqrt2xy)$).
– metamorphy
Aug 12 at 14:28
















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