= Solution
Let $d=b-a$. In the <mixed Dirichlet-Neumann problem in an elliptic annulus>, the angular <Fourier modes> separate. The outer data excite the $\cos\nu$ mode, while the inner derivative excites $\cos2\nu$.
For the first mode, a radial factor with zero derivative at $\mu=a$ is $\cosh(\mu-a)$. Normalizing its outer value gives $F_1(\mu)=\cosh(\mu-a)/\cosh d$. For the second mode, a radial factor vanishing at $\mu=b$ is $\sinh[2(b-\mu)]$. Its derivative at $a$ must be $-1$, fixing the coefficient to $1/[2\cosh(2d)]$. Therefore
$$
\boxed{\Phi(\mu,\nu)=\frac{\cosh(\mu-a)}{\cosh(b-a)}\cos\nu+\frac{\sinh[2(b-\mu)]}{2\cosh[2(b-a)]}\cos2\nu.}
$$
Each term satisfies <Laplace's equation>; at $\mu=b$ the second term vanishes and the first is $\cos\nu$. At $\mu=a$, differentiating gives $0-\cos2\nu$, as required. All other modes, including the constant mode, vanish by the homogeneous mixed conditions. Alternatively, <integration by parts> applied to the difference of two solutions gives a zero integral of $|\nabla\Phi|^2$, proving <uniqueness>.
\b[Boundary convention.] The calculation follows the explicitly printed coordinate derivative $\partial_\mu\Phi$. In <elliptic coordinates>, the <scale factor> is $h=\sqrt{\sinh^2\mu+\sin^2\nu}$ and the outward <unit normal> at the inner boundary points toward decreasing $\mu$. A physical <normal derivative> there would be $-h^{-1}\partial_\mu\Phi$. Thus the phrase “normal derivative” in the source must be understood in its displayed coordinate sense; imposing the same numerical data on a unit-normal derivative would be a different problem.
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