Question Number 137275 by mnjuly1970 last updated on 31/Mar/21
$$\:\:\:\:\:\:\:\:\:\:\:\:\:……{complex}\:\:{analysis}….. \\ $$$$\:\:\:\:{if}\:,\:\:{f}\left(\alpha,{n},{x}\right)=\frac{{d}^{\:{n}} }{{dx}^{{n}} }\left(\alpha^{{x}} \right)\:\:,\:{x}\in\mathbb{C} \\ $$$$\:\:\:\:\:\alpha\in\mathbb{C}−\left\{\mathrm{0}\right\}\:,\:{n}\in\mathbb{C}−\mathbb{Z}^{−} \cup\left\{\mathrm{0}\right\} \\ $$$$\:\:\:\:\:{and}\:\:{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {f}\left(\alpha,{n},{x}\right){d}\alpha \\ $$$$\:\:\:\:\:{then}\:\:{find}\:\:{the}\:{value}\:{of}\:… \\ $$$$\:\:\:\:\:\:\Omega=\frac{{Im}\left({g}\left({i},\mathrm{0}\right)\right)}{{Re}\left(\Gamma\left({i}\right)\right)} \\ $$$$\:\:\:\:\:\:{solution}: \\ $$$$\:\:\:\:\:\:{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{d}^{\:{n}} }{{dx}^{{n}\:} }\left(\alpha^{{x}} \right){d}\alpha \\ $$$$\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{d}^{\:{n}} }{{dx}^{{n}} }\left({e}^{{xln}\left(\alpha\right)} \right){d}\alpha\:…..\langle\ast\rangle \\ $$$$\:\:\:\:\:\:\:\frac{{d}^{\:{n}} }{{dx}^{{n}} }\left({e}^{{xln}\left(\alpha\right)} \right)=\left({ln}\left(\alpha\right)\right)^{{n}} \alpha^{{x}} \\ $$$$\:\:\:\langle\ast\rangle\rightarrow\:…{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({ln}\left(\alpha\right)\right)^{{n}} \alpha^{{x}} {d}\alpha \\ $$$$\:\:\underset{\alpha={e}^{−{y}} } {\overset{{ln}\left(\alpha\right)=−{y}} {=}}\int_{\mathrm{0}} ^{\:\infty} \left(−\mathrm{1}\right)^{{n}} {y}^{\:{n}} {e}^{−{yx}} {e}^{−{y}} {dy} \\ $$$$\:\:\:\:\:=\left(−\mathrm{1}\right)^{{n}} \int_{\mathrm{0}} ^{\:\mathrm{1}} {y}^{{n}} .{e}^{−{y}\left(\mathrm{1}+{x}\right)} {dy} \\ $$$$\:\:\:\:\overset{{y}\left(\mathrm{1}+{x}\right)={t}} {=}\left(−\mathrm{1}\right)^{{n}} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{t}^{{n}} \:{e}^{−{t}} }{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }{dt} \\ $$$$\:\:\:\:\:\:\:={e}^{{in}\pi} .\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }\:.\Gamma\left({n}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:{g}\left({i},\mathrm{0}\right)={e}^{−\pi} .{i}.\Gamma\left({i}\right)\:……\langle\ast\ast\rangle \\ $$$$\:\:\:\Gamma\left({i}\right)\in\mathbb{C}\:\Rightarrow\:\:\Gamma\left({i}\right)={Re}\left(\Gamma\left({i}\right)\right)+{Im}\left(\Gamma\left({i}\right)\right) \\ $$$$\:\:\:\langle\ast\ast\rangle\rightarrow\:…\:{g}\left({i},\mathrm{0}\right)={e}^{−\pi} .{i}.\left[{Re}\left(\Gamma\left({i}\right)\right)+{Im}\left(\Gamma\left({i}\right)\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\Omega=\frac{{e}^{−\pi} {Re}\left(\Gamma\left({i}\right)\right)}{{Re}\left(\Gamma\left({i}\right)\right)}\:={e}^{−\pi} …\checkmark\checkmark \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$