I am trying to prove that $$ \frac{1}{2\pi i}\int^{a+i\infty}_{a-i\infty}\frac{x^s}{s-\beta}ds =\begin{cases} x^{\beta}, & x > 1 \\ 0, & 0 < x < 1 \\ \end{cases} $$ for $0<{\rm Re}(\beta)<a$ by using Feynman integration and solving a homogeneous second order differential equation respectively. Here is my attempt: \begin{align*} \frac{1}{2\pi i}\int^{a+i\infty}_{a-i\infty}\frac{x^s}{s}ds& =\frac{x^a}{2\pi}\int^{\infty}_{0}\frac{x^{it}(a-it)+x^{-it}(a+it)}{a^2+t^2}dt\\ & =\frac{x^a}{\pi}\left(\int^{\infty}_{0}\frac{\cos(\ln(x)ta)}{1+t^2}dt+\int^{\infty}_{0}\frac{t\sin(\ln(x)ta)}{1+t^2}dt\right). \end{align*} \begin{align*} {\rm Re}(\oint_C\frac{x^{iza}}{1+z^2}dz) & ={\rm Re}\left(\lim_{R \to \infty}\int^{R}_{-R}\frac{x^{iza}}{1+z^2}dz+\int_{\Gamma}\right) \\ & =\frac{\pi}{x^a}.\lim_{R \to \infty} \left| \int_{\Gamma} \right| \\ & \leqslant\lim_{R \to \infty} \frac{2}{R}\int^{\frac{\pi}{2}}_{0}\frac{x^{-2R\theta/\pi}}{1+(Re^{i\theta})^{-2}}d\theta \\ & =0. \end{align*} Therefore $$ \int^{\infty}_{0}\frac{\cos(\ln(x)ta)}{1+t^2}dt = \frac{\pi}{2x^a}. $$ Set $$ I(a):=\int_{R}\frac{\cos(\ln(x)ta)}{1+t^2}dt $$ so $$ I'(a)=\ln(x)(-{\rm sgn}(\ln(x))\pi+\int_{R}\frac{\sin(\ln(x)ta)}{t(1+t^2)}dt) $$ and $$ I''(a)-\ln^2(x)I(a) = 0. $$ Therefore $I(a)=c_1x^a+c_2x^{-a}$ for $$ c_1=\begin{cases} 0, & x > 1 \\ \pi, & 0 < x < 1 \\ \end{cases} \text{ and } c_2= \begin{cases} \pi, & x > 1 \\ 0, & 0 < x < 1.\end{cases} $$
Assuming my attempt is correct thus far, can I evaluate $$ \int_{R}\frac{t\sin(\ln(x)at)}{1+t^2}dt=-\frac{I'(a)}{\ln(x)}:=D(a)={\rm sgn}(\ln(x))\pi-\int_{R}\frac{\sin(\ln(x)at)}{t(1+t^2)}dt. $$ Then $D'(a)=-\ln(x)I(a)$, so $$D(a)=c_2x^{-a}-c_1x^{a}.$$ $$ \frac{1}{2\pi i}\int^{a+i\infty}_{a-i\infty}\frac{x^s}{s}ds=\frac{x^{a}}{2\pi}((c_1x^{a}+c_2x^{-a})+c_2x^{-a}-c_1x^{a})=\begin{cases} 1, & x > 1 \\ 0, & 0 < x < 1. \end{cases} $$ (as defined herein) using Feynman integration? It is used extensively in Riemann's paper in finding an analytic representation of the jump function (equivalently, an asymptotic estimate of the prime counting function by applying the Moebius inversion formula [following from the associativity of the Dirichlet convolution] to the analytic representation of the jump function) and in a proof of Perron's formula. Pardon my illegible uglyography and quotidian rogitation, albeit I consider the transient apanthropinization as a pars pro toto for expunged pedagogical anomalies.
