Question Number 28676 by abdo imad last updated on 28/Jan/18
$${let}\:{give}\:\:{u}_{{n}} =\:\int_{{n}\pi} ^{\left({n}+\mathrm{1}\right)\pi} \:\:{e}^{−\lambda{t}} \:\frac{{sint}}{\sqrt{{t}}}\:\:\:\:{with}\:\lambda>\mathrm{0} \\ $$$${calculate}\:\sum_{{n}=\mathrm{0}} ^{+\infty} \:\:\:{u}_{{n}} \:.\: \\ $$$$ \\ $$
Commented by abdo imad last updated on 30/Jan/18
$$\sum_{{n}=\mathrm{0}} ^{+\infty} \:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{t}} \:\frac{{sint}}{\sqrt{{t}}}{dt}=\sqrt{\pi}{sin}\left(\:\frac{\pi}{\mathrm{4}}\:−\frac{{arctan}\lambda}{\mathrm{2}}\right)\:{from} \\ $$$${Q}\:\mathrm{28756}. \\ $$