Do analog circuits exist that are essentially mathematical functions in DC?
Analog Devices have papers on the subject, e.g. this one on log amps.

That paper links to the AD538:


  The AD538 is a monolithic real-time computational circuit which provides precision analog multiplication, division, and exponentiation. The combination of low input and output offset voltages and excellent linearity results in accurate computation over an unusually wide input dynamic range. Laser wafer trimming makes multiplication and division with errors as low as 0.25% of reading possible, while typical output offsets of 100 microV or less add to the overall off-the-shelf performance level. Real-time analog signal processing is further enhanced by the device's 400 kHz bandwidth.


It's important to note those limitations. 400kHz bandwidth is not very high compared to doing the same arithmetic in the digital domain. That's one of the reasons why almost all control systems computation is done in digital; the others are better linearity, noise tolerance, thermal stability and power consumption.