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Question Number 144909 by ArielVyny last updated on 30/Jun/21

$$\mathrm{\Gamma }(a+ib){doesn}^{\prime }texist?givehervalue$$

Answered by qaz last updated on 30/Jun/21

$$\mathrm{\Gamma }(\mathrm{a}+\mathrm{ib})={\int }_0^{\mathrm{\infty }}{\mathrm{x}}^{\mathrm{a}+\mathrm{ib}-1}{\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}$$$$={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{iblnx}}\cdot {\mathrm{x}}^{\mathrm{a}-1}{\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}$$$$={\int }_0^{\mathrm{\infty }}(\cos \left(\mathrm{blnx}\right)+\mathrm{isin}\left(\mathrm{blnx}\right)){\mathrm{x}}^{\mathrm{a}-1}{\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}$$$$={\int }_0^{\mathrm{\infty }}\cos \left(\mathrm{blnx}\right)\cdot {\mathrm{x}}^{\mathrm{a}-1}{\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}+\mathrm{i}{\int }_0^{\mathrm{\infty }}\sin \left(\mathrm{blnx}\right)\cdot {\mathrm{x}}^{\mathrm{a}-1}{\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}$$

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