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The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\qquad\qquad(\ast)$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if $f(x)$ has compact support. A corollary is $$ \lim_{\epsilon\to 0^+}\int_a^b \frac{\epsilon f(x)}{[(b-x)(x-a)]^{1-\epsilon}}\,dx=\frac{f(a)+f(b)}{b-a}.$$

All of this can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)}{x^{1-\epsilon}}=\delta(x),\qquad\qquad(\ast\ast)$$ acting on compactly supported functions.

Q: One can readily check the formula $(\ast)$ for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation of the delta function known?

The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\label{1}\tag{$\ast$}$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if $f(x)$ has compact support. A corollary is $$ \lim_{\epsilon\to 0^+}\int_a^b \frac{\epsilon f(x)}{[(b-x)(x-a)]^{1-\epsilon}}\,dx=\frac{f(a)+f(b)}{b-a}.$$

All of this can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)}{x^{1-\epsilon}}=\delta(x),\tag{$\ast\ast$}$$ acting on compactly supported functions.

Q: One can readily check the formula \eqref{1} for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation of the delta function known?

The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\qquad\qquad(\ast)$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if $f(x)$ has compact support. This can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)}{x^{1-\epsilon}}=\delta(x),\qquad\qquad(\ast\ast)$$ acting on compactly supported functions.

Q: One can readily check the formula $(\ast)$ for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation of the delta function known?

The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\qquad\qquad(\ast)$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if $f(x)$ has compact support. A corollary is $$ \lim_{\epsilon\to 0^+}\int_a^b \frac{\epsilon f(x)}{[(b-x)(x-a)]^{1-\epsilon}}\,dx=\frac{f(a)+f(b)}{b-a}.$$

All of this can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)}{x^{1-\epsilon}}=\delta(x),\qquad\qquad(\ast\ast)$$ acting on compactly supported functions.

Q: One can readily check the formula $(\ast)$ for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation of the delta function known?

The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dx=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\qquad\qquad(\ast)$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if the integral converges.

It can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)\theta(b-x)}{x^{1-\epsilon}}=\delta(x).$$

Q: One can readily check the formula $(\ast)$ for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation known?

The Dirac delta function appears in the Sokhotsky formula, $$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$ to be understood in the integral sense $$\text{Im}\lim_{\epsilon\to 0^+} \int \frac{f(y)}{y-x-i\epsilon}dy=\pi f(x),$$ for a real valued function $f(x)$.

I stumbled on an identity that has a similar flavour, $$\lim_{\epsilon\to 0^+}\int_x^b \frac{\epsilon f(y)}{(y-x)^{1-\epsilon}} dy=f(x).\qquad\qquad(\ast)$$ The upper limit $b>x$ of the integral is arbitrary, one may send it to infinity if $f(x)$ has compact support. This can be interpreted as a delta function representation in terms of the unit step function $\theta(x)$, $$\lim_{\epsilon\to 0^+} \frac{\epsilon\theta(x)}{x^{1-\epsilon}}=\delta(x),\qquad\qquad(\ast\ast)$$ acting on compactly supported functions.

Q: One can readily check the formula $(\ast)$ for polynomial functions $f(x)$. Is there a more comprehensive derivation? Is this representation of the delta function known?