Function integral on ℝⁿ
Set
context | p∈N |
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 x∈Rp of f becoming a dummy index.
Theorems
For f:X→R…differentiable and f′…bounded, we have
∫badfdxdx=f(b)−f(a) |
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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]:=1b−a∫baf(x)dx
f(a)+f(b)2≥⟨f⟩[a,b]≥f(a+b2) |
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See references.
Kernel of he integral
A linear combination of functions that are zero under an integral are again zero.
Special case
∫a−aE(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:
∫a−aE(x)11±eU(x)dx=∫a0E(x)dx
∫a−af(x2)11+ex2sin(x)dx=∫a0f(x2)dx
References
Wikipedia: Hermite–Hadamard inequality