= Solution
Let $S'$ move at $+v$ along $x$ relative to $S$. The <Lorentz transformation> of <energy> and <momentum>, from primed to unprimed components, is
$$
E=\gamma(E'+vp_x'),\qquad
p_x=\gamma\left(p_x'+\frac{vE'}{c^2}\right),\qquad
p_y=p_y',\qquad p_z=p_z'.
$$
The inverse equations have $v$ replaced by $-v$ and the primes interchanged. A <photon> making angle $\theta'$ with $+x$ in $S'$ has $p_x'=(E'/c)\cos\theta'$. Thus its general <Doppler factor> is
$$
D=\frac E{E'}=\gamma(1+\beta\cos\theta').
$$
The requested blueshifted expression assumes propagation parallel to the boost, $\theta'=0$:
$$
\boxed{D=\gamma(1+\beta)=\sqrt{\frac{1+\beta}{1-\beta}}.}
$$
The propagation-direction assumption matters: for an oppositely travelling <photon>, $D=\gamma(1-\beta)$, the reciprocal of the displayed factor. Relative frame <speed> alone does not select the longitudinal <relativistic Doppler effect>.
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