- #1
luisgml_2000
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Hello!
I'm studying on my own the complex error function w(z), also known as Faddeyeva function. On page 297 from Abramowitz it is stated that
$$
\frac{i}{\pi} \int_0^{\infty} \frac{e^{-t^2}}{z-t}\, dt=e^{-z^2}\operatorname{erfc}(-iz)
$$
where
<br /> \operatorname{erfc}(z)=\frac{2}{\sqrt{\pi}}\int_z^\infty e^{-t^2} \, dt<br />
The former identity is puzzling me and therefore I can't come up with a proof for it. Welcome any suggestions!
Thanks in advance for your attention.
I'm studying on my own the complex error function w(z), also known as Faddeyeva function. On page 297 from Abramowitz it is stated that
$$
\frac{i}{\pi} \int_0^{\infty} \frac{e^{-t^2}}{z-t}\, dt=e^{-z^2}\operatorname{erfc}(-iz)
$$
where
<br /> \operatorname{erfc}(z)=\frac{2}{\sqrt{\pi}}\int_z^\infty e^{-t^2} \, dt<br />
The former identity is puzzling me and therefore I can't come up with a proof for it. Welcome any suggestions!
Thanks in advance for your attention.
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