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On Alonso Finn I found the following formula while studying the Compton effect, which should show that the relativistic relation between kinetic energy of electron $E_k$ and electron momentum $p_e$ can be approximated in the following way:

$$E_k=c \sqrt{m_e^2 c^2+p_e^2 }-m_e c^2 \approx \frac{p_e^2}{2 m_e}.\tag{1}$$

Is this correct? I do not get it, since it should be

$$c \sqrt{m_e^2 c^2+p_e^2 }-m_e c^2=m_e c^2 \left(\sqrt{1+(\frac{p_e}{m_e c})^2}-1\right)\approx \frac{p_e}{2m_e c}.\tag{2}$$

And this hold if $p_e<<m_e c$ (which means that $v_e<< c$?)

How is $(1)$ actually derived correctly?

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$$m_e c^2\left(\sqrt{1+\left(\frac{p_e}{m_e c}\right)^2}-1\right)\approx m_ec^2\left(1+\frac{1}{2}\left(\frac{p_e}{m_e c}\right)^2-1\right)$$ $$=m_e c^2\frac{p_e^2}{2m_e^2 c^2}=\frac{p_e^2}{2m_e}$$

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