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Function integral on ℝⁿ

Set

context pN
definiendum Ip:(Rp¯R)¯R
definiendum Ip(f):=Rp f dλp

Discussion

Because the integral above coincides with the Lebesgue–Stieltjes integral for the monotone function F(x):=x, we'll also denote Ip(f) by Rp f(x) dxp with the argument xRp of f becoming a dummy index.

Theorems

For f:XR…differentiable and fbounded, we have

badfdxdx=f(b)f(a)
d(w(y)v(y)f(x)dx)=f(v(y))dv(y)f(w(y))dw(y)

For f convex and

f[a,b]:=1babaf(x)dx

f(a)+f(b)2f[a,b]f(a+b2)

See references.

Kernel of he integral

A linear combination of functions that are zero under an integral are again zero.

Special case

aaE(x)(12+k=0ckUk(x)2k+1)dx=a0E(x)dx

e.g. all Uk the same and ck so that you get 11±ey:

aaE(x)11±eU(x)dx=a0E(x)dx

aaf(x2)11+ex2sin(x)dx=a0f(x2)dx

References

Subset of

Context

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