APCMLC, the apodizer function is chosen equal to $\psi_a$:

$$\Psi_A = \psi_a P$$

The focal plane mask complex amplitude is :

$$F_{mask} = 1 - (1-t) M_a$$

where $t$ is the complex amplitude transmission within the circular focal plane mask. The complex amplitude in plane B is :

$$\Psi_B = F_{mask} \widetilde{\Psi_A} = \widetilde{\Psi_A} - (1-t) M_a \widetilde{\Psi_A}$$

The complex amplitude in plane C is obtained by Fourier transform of $\Psi_B$: $$\label{equ:PsiC0}
\Psi_C = \Psi_A - (1-t) (\psi_a P) \otimes \widetilde{M_a}$$

By combining equations \[equ:La\] and \[equ:PsiC0\], and multiplying by $P(r)$, the complex amplitude in plane C within the geometrical pupil is:

$$\label{equ:PsiC}
\Psi_C P(\mathbf{r}) = P(\mathbf{r}) \times (1- (1-t) \Lambda_a) \psi_a$$

This equation shows that, if $t = 1-\Lambda_a^{-1}$ (this value is now noted $t_a$), then $\Psi_C$ is equal to zero within the geometric pupil. This is the condition for a APCMLC, which completely removes light from an on-axis point source, provided that a Lyot pupil plane mask $L_{mask}(\mathbf{r})=P(\mathbf{r})$ is used to only select light within the geometric pupil. Since $\Lambda_a < 1$, $t_a$ is negative: the focal plane mask is both partially transmissive and π-phase shifting. A coronagraphic solution requires $t_a>-1$, and therefore exists only if $\Lambda_a > 