Electronic – Can digital clocks and other electronic devices “ride through” a momentary power interruption

Energy in the storage device is \$\frac{1}{2}CV^2\$, or \$4.7 \times 10^{27}\$ Joules. Energy consumed per activation is \$1.21 \times 10^{9}\$ Joules. So you can get ~\$3.89 \times 10^{18}\$ activations, or nine cycles every second. Every second. All of them.

At first I thought that to get that much energy, you'd have to consume the universe. Then I thought the destruction might in fact be very localized, limited to merely our own galaxy. But that squared term in \$E=Mc^2\$ really dominates! We're only talking about annihilating 52 gigagrams, or the mass of the Titanic. All the energy stored in your capacitor, giving that impressively large number of activations, is comparable to the power output of the sun in a tenth of a second.

So I don't know why everyone's saying this is unreasonable!

There is no real difference between an analogue power supply and a digital power supply. They are separated because the digital power supply will supply current to digital switches which will switch on and off rapidly, putting large current demand spikes onto the power line, and hence cause momentary voltage drops and spikes on the line - these spikes are managed by decoupling capacitors local to the switching devices, and bulk capacitance at the power supply.

By keeping the two supply lines separate the circuit designer can isolate these current spikes from analogue circuitry which may be affected by small voltage drops - for example if the supply is used as a reference for an A2D. Often the two supplies are fed from the same source, but the analogue supply may have a small RC or inductive filter in line to remove digital noise present at the source.

So yes, the 3.3v pin of the STM32F4-discovery board could be used as an analog power source. I would just add a small series resistor (less than 0.05V drop at the current you are using) and additional capacitance to ground - to give a first order low pass filter frequency of about 1KHz - or whatever you deem appropriate.