Power of an angle-modulated wave Let $$\phi(t) = A\cos(\omega_c t + km(t)) \tag{1}$$ be an angle-modulated wave(FM or PM). What's the power of $\phi(t)$? Intuitively, it seems that the answer is $$P = \frac{A^2}{2} \tag{2}$$ since the amplitude $A$ remains constant but I couldn't prove that using the definition which gives $$\lim_{T \to +\infty} \frac{1}{T}\int_{-\frac{T}2}^{\frac{T}2} A^2\cos^2(\omega_ct + km(t))dt \tag{3}$$ It seems we can't evaluate $(3)$ in the general case $m(t)$.
Let's consider the complex signal
$$\phi_c(t)=Ae^{j(\omega_ct+km(t))}\tag{1}$$
The power of $(1)$ is readily computed as
$$P_{\phi_c}=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\big|\phi_c(t)\big|^2dt=A^2\tag{2}$$
IF $(1)$ were the analytic signal corresponding to the real-valued signal $\phi(t)$ then the power of $\phi(t)$ would simply be $P_{\phi_c}/2=A^2/2$.
In general $(1)$ is not analytic. However, in all practical cases of angle modulation, a sufficiently high carrier frequency makes $(1)$ analytic for all practical purposes, i.e., there are no measurable negative frequency components. Or, equivalently, a practical angle modulated signal is a bandpass signal and has no components at or close to DC. If this is the case, the power of $\phi(t)$ is indeed given by $A^2/2$.
In all other (artificial) cases, the power of $\phi(t)$ cannot generally be shown to equal $A^2/2$.