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let-w-e-ipi-4-1-i-2-show-that-1-1-i-erf-wx-pi-2-0-x-e-i-t-2-pi-2-dt-c-x-is-x-




Question Number 173116 by ali009 last updated on 06/Jul/22
let w=e^(iπ/4) =(1+i)/(√(2 )) show that:  (1/(1+i))erf(wx(√(π/2)))=∫_0 ^x e^(−i t^2  π/2)  dt=c(x)−is(x)
$${let}\:{w}={e}^{{i}\pi/\mathrm{4}} =\left(\mathrm{1}+{i}\right)/\sqrt{\mathrm{2}\:}\:{show}\:{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+{i}}{erf}\left({wx}\sqrt{\frac{\pi}{\mathrm{2}}}\right)=\int_{\mathrm{0}} ^{{x}} {e}^{−{i}\:{t}^{\mathrm{2}} \:\pi/\mathrm{2}} \:{dt}={c}\left({x}\right)−{is}\left({x}\right) \\ $$
Commented by ali009 last updated on 06/Jul/22
note/ c(x)and s(x) are fresnel integrals
$${note}/\:{c}\left({x}\right){and}\:{s}\left({x}\right)\:{are}\:{fresnel}\:{integrals} \\ $$

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