  We studied single $W$ boson production in high energy $e\gamma$ collisions
and the sensitivity of various observables to the $WW\gamma$ gauge boson
coupling by evaluating the helicity amplitudes of all Feynman diagrams which
contribute to the final state being studied, including the $W$ decay to final
state fermions. We examined $W$ production at 500 GeV and 1 TeV $e^+e^-$
colliders, comparing results for photon spectra obtained from a backscattered
laser and from beamstrahlung radiation. Here we found that the couplings could
best be measured using the backscattered laser photons with
$|\delta\kappa_\gamma|\leq 0.07$ and $|\lambda_\gamma| \leq 0.05$ at a 500 GeV
collider and $|\delta\kappa_\gamma|\leq 0.07$ and $|\lambda_\gamma| \leq 0.02$
at a 1 TeV collider, all at 95% C.L.. The measurement of $\kappa_\gamma$ is at
the threshold of being able to measure loop contributions to the trilinear
gauge boson vertex. For completeness we include the limits achievable using
single $W$ production at a 200 GeV $e^+e^-$ collider in the Weizacker-Williams
approximation. This process can measure $\kappa_\gamma$ to $\pm 0.15$ at 95%
C.L. which is comparable to the $W$ pair production process.
