Question Number 67537 by mathmax by abdo last updated on 28/Aug/19 $${prove}\:{that}\:\frac{\mathrm{1}}{\Gamma\left({z}\right)}\:={z}\:{e}^{\gamma{z}} \:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{{z}}{{n}}\right){e}^{−\frac{{z}}{{n}}} \\ $$ Commented by ~ À ® @ 237 ~…
Question Number 67534 by mathmax by abdo last updated on 28/Aug/19 $${find}\:{the}\:{value}\:{of}\:\:\prod_{{n}=\mathrm{2}} ^{\infty} \:\frac{{n}^{\mathrm{3}} −\mathrm{1}}{{n}^{\mathrm{3}} \:+\mathrm{1}} \\ $$$${and}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right) \\ $$ Commented by…
Question Number 67532 by mathmax by abdo last updated on 28/Aug/19 $${prove}\:{that}\:\:\pi\:{cotan}\left(\pi\alpha\right)\:={lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{{k}=−{n}} ^{{n}} \:\:\frac{\mathrm{1}}{\alpha−{k}} \\ $$ Commented by ~ À ® @ 237 ~…
Question Number 67533 by mathmax by abdo last updated on 28/Aug/19 $${find}\:{the}\:{value}\:{of}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}\right) \\ $$$$\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{\mathrm{2}}{{n}\left({n}+\mathrm{1}\right)}\right) \\ $$ Terms of Service Privacy Policy…
Question Number 67524 by mathmax by abdo last updated on 28/Aug/19 $${prove}\:{that}\:\forall{z}\:\in{C}\:\:{we}\:{have} \\ $$$${sinz}\:={z}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{z}^{\mathrm{2}} }{{n}^{\mathrm{2}} \pi^{\mathrm{2}} }\right) \\ $$ Commented by ~ À…
Question Number 67522 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{z}\:{from}\:{C}−{Z}\:\:\:\:\:{prove}\:{that} \\ $$$$\frac{\pi}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} }\:\:{and} \\ $$$$\frac{\pi{cos}\left(\pi{z}\right)}{{sin}\left(\pi{z}\right)}\:=\frac{\mathrm{1}}{{z}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{2}{z}}{{z}^{\mathrm{2}} −{n}^{\mathrm{2}} }…
Question Number 67520 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{f}\left({x},{z}\right)\:=\frac{{z}\:{e}^{{xz}} }{{e}^{{z}} −\mathrm{1}}\:\:\:\:\:\:\left({x}\:{and}\:{z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\left({x},{z}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \left({x}\right)\frac{{z}^{{n}} }{{n}!} \\ $$$${with}\:{B}_{{n}} \left({x}\right)\:{is}\:{a}\:{unitaire}\:{polynome}\:{with}\:{degre}\:{n} \\ $$$${determine}\:{B}_{{n}}…
Question Number 67521 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\:\:{A}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{cos}\left({nx}\right)}{{n}} \\ $$$${and}\:{B}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \:{sin}\left({nx}\right)}{{n}} \\ $$ Terms of Service…
Question Number 67518 by mathmax by abdo last updated on 28/Aug/19 $${if}\:{z}\:={x}+{iy}\:\:\:{find}\:\:{lnz}\:\:{interms}\:{of}\:{x}\:{and}\:{y} \\ $$$$ \\ $$ Commented by mathmax by abdo last updated on 30/Aug/19…
Question Number 67519 by mathmax by abdo last updated on 28/Aug/19 $${if}\:\frac{{z}}{{e}^{{z}} −\mathrm{1}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \:\frac{{z}^{{n}} }{{n}!} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{B}_{\mathrm{0}} ,{B}_{\mathrm{1}} ,{B}_{\mathrm{2}} ,{B}_{\mathrm{3}} ,{B}_{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{z}\rightarrow\frac{\mathrm{1}}{{e}^{{z}}…