= Solution
Boundary-layer theory assumes steady incompressible high-Reynolds-number flow, $\delta/x\ll1$, weak streamwise viscous diffusion, nearly constant <pressure> across the layer, and outer <pressure> <gradient> $-\rho^{-1}p_x=U\,U'$.
For $\phi=-A r^k\cos(k\theta)/k$,
$$
u_r=-Ar^{k-1}\cos k\theta,\qquad u_\theta=Ar^{k-1}\sin k\theta.
$$
At $\theta=\pm\pi/k$ the normal <velocity> vanishes; along the upper wall $U(x)=Ax^{k-1}$. <Boundary-layer scaling> gives
$$
\delta=\sqrt{\frac{\nu x}{U}}=(\nu x^{2-k}/A)^{1/2}.
$$
Set $m=k-1$. The <Falkner-Skan equation> becomes
$$
f^{(3)}=(k-1)f'^2-\frac{k}{2}ff''+(1-k),
$$
so $\alpha=k-1$, $\beta=-k/2$, $\gamma=1-k$, with
$f(0)=f'(0)=0$ and $f'(\infty)=1$. Finally
$$
\frac{\delta}{x}=(\nu/A)^{1/2}x^{-k/2}\ll1
$$
requires $x\gg(\nu/A)^{1/k}$; thus $a=1/k$ and the tip region violates the approximation.
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