1. **State the problem:** We want to find the intensity $|B|^2$ of the emerging wave from a Fabry–Pérot interferometer, where the wave amplitude is given by $$B = A(1-r) \sum_{n=0}^\infty r^n e^{i n \phi},$$ with $r$ and $\phi$ real numbers, and $A$ possibly complex. 2. **Recognize the series:** The sum is a geometric series with common ratio $r e^{i \phi}$. 3. **Sum the geometric series:** For $|r e^{i \phi}| = r < 1$, the sum converges to $$\sum_{n=0}^\infty r^n e^{i n \phi} = \frac{1}{1 - r e^{i \phi}}.$$ 4. **Substitute back into $B$:** $$B = A(1-r) \cdot \frac{1}{1 - r e^{i \phi}} = \frac{A(1-r)}{1 - r e^{i \phi}}.$$ 5. **Find the intensity $|B|^2$:** Intensity is the squared magnitude of $B$: $$|B|^2 = B \cdot B^* = \left| \frac{A(1-r)}{1 - r e^{i \phi}} \right|^2 = \frac{|A|^2 (1-r)^2}{|1 - r e^{i \phi}|^2}.$$ 6. **Calculate the denominator magnitude:** $$|1 - r e^{i \phi}|^2 = (1 - r e^{i \phi})(1 - r e^{-i \phi}) = 1 - r e^{i \phi} - r e^{-i \phi} + r^2.$$ 7. **Simplify using Euler's formula:** Since $e^{i \phi} + e^{-i \phi} = 2 \cos \phi$, we get $$|1 - r e^{i \phi}|^2 = 1 - 2 r \cos \phi + r^2.$$ 8. **Final expression for intensity:** $$|B|^2 = \frac{|A|^2 (1-r)^2}{1 - 2 r \cos \phi + r^2}.$$ This formula gives the intensity of the emerging wave in terms of $A$, $r$, and $\phi$.