First, let's go over what an analog computer is. An analog computer, put simply, is a device that accepts input parameters which are represented by something inside the computer. It performs computing functions through mechanisms and/or electronic circuits, and the outputs are expressed by quantities of something the mechanism can produce.
A Philbrick K-3 analog computer, circa 1950. From the Philbrick Archives.
Analog computers preceded the development of electricity. The first, simple analog computing devices go back to the Middle Ages, but significant ones started appearing during the pre-industrial scientific discovery period from 1600 to 1800. Generally they relied on sliding or rotating parts to represent measurements which were input or output. A simple but important example is the slide rule, invented in the 17th century. A basic slide rule multiplies two numbers by positioning one operand on a sliding scale against a fixed scale; the amount by which the sliding scale is moved represents (by reading off of a scale) the product.
In the early 20th century, a number of powered analog computers were invented to do specific calculations. An early driver behind the development of this technology was the need for a device called a "gun director". This was a computer that computed the elevation and azimuth angles at which an artillery piece needed to be pointed in order to hit a target, given the range to the target, the wind, the weight of the shell being fired, and possibly other factors. The Norden bombsight was a famous electro-mechanical analog computer deployed by the Allies during World War II. To use it, a bombardier looked through a sight glass to find the target to be bombed. From the pointing angles of the sight, and the rate at which the bombardier had to move the sight in order to keep it on target, the bombsight computed the heading that the aircraft needed to fly, and the time at which the bombs should be dropped. In this computer, the quantities being computed were represented by the movements of levers or gears. (The bombsight was usually coupled to the airplane's autopilot so it could actually fly the aircraft during the bombing run, and to the bomb racks so it could release the bombs at the right time automatically.)
Norden bombsight (top left) and servos controlled by the bombsight.
Electronic analog computers started to appear around 1930. As was generally the case of the mechanical analog computers, most of the early electronic devices were hard-wired to perform a specific computation; because of this, early uses were limited to problems that were both important and difficult, enough that it was worth the cost to build a computer. An early example was a device known as the "AC network analyzer", which was built to solve problems that electrical power utilities were encountering as individual power stations were being combined into large grids.
In 1938, electrical engineer George A. Philbrick, then employed by the Foxboro Company of Massachusetts, wrote a proposal for an electronic analog computer that would model various types of closed-loop manufacturing processes. One of the problems that Philbrick had to solve was how to design circuits that would perform the needed math operations in a general sense, that is, not specific to a particular problem. In 1943, Philbrick was working on a contract with the U.S. Army to devise improvements to the M9 gun director, which had been built by Bell Labs. It worked, but it was too slow to compute in real time. Philbrick came in contact with Loeb Julie of Columbia University, who had devised the first experimental operational amplifiers. (Yes, there were op amps decades before the first integrated circuits.) Philbrick realized that Julie's op amps could be used to perform a variety of analog computing math functions, and he began working on his own improvements.
Philbrick K2-P op amp
After WWII, Philbrick started his own company, George A. Philbrick Researches. The company was heavily involved in both analog computing and commercial op amp design and manufacturing. The company published a widely regarded collection of papers and notes concerning analog computing -- system design, circuit design, programming, and operations. Analog computers were becoming more compact, and general-purpose units were appearing that offered a number of function modules which could be interconnected by the user in any desired configuration using patch cords. In fact, Philbrick's company developed the idea of a "modular computer", in which individual function blocks could be purchased and combined as needed to apply to a problem -- a concept very similar to the modular computers that would come later. At some point Philbrick hired a certain young electrical engineer, one Alan R. Pearlman, who took an interest in the op amp business. So much so that, in the early 1960s, Pearlman and another Philbrick employee broke away and established their own company, Nexus Research Labs, which continued their work in the op-amp and analog computing business.
Philbrick K3 analog computer modules. From the Philbrick Archive.
If Pearlman's name doesn't sound familar, look at his initials -- A.R.P. In 1966, Pearlman's group sold Nexus Research Labs to Teledyne. The sale made Pearlman wealthy, and he used some of that wealth to found ARP Instruments. Look at the photo above. Looks vaguely familiar? The Philbrick analog computer systems were modular. There were about 10 function modular that the user could purchase and configure in a case as needed. Compare to this:
ARP 2500 model 1947 voltage controlled filter
However... The first of what we consider synthesizers today didn't come from Pearlman. The two men who are generally credited with developing the basic building blocks of the analog synthesizer -- the voltage controlled oscillator, filter, and amplifier -- are Robert Moog and Don Buchla. Moog has an obvious, if indirect, connect to Philbrick via Columbia University, where Philbrick and Loeb Julie worked on the first op-amp designs in the 1940s, and where John Ragazzini and Rudolf Kalman had continued to work on analog computing concepts through the 1950s. The Columbia-Princeton Electronic Music Center opened at Columbia in the mid-1950s, but it is not clear how much cross-fertilization there was between it and the analog computing labs. Moog just missed experiencing the RCA Synthesizer, which was installed at the center in 1958; he had graduated in '57.
Little is written about what Moog actually studied or did at Columbia (far more is written about his theremin side business by which he paid his way through school), so further investigation is difficult. He did get his degree there in electrical engineering, and in a mid-1950s electrical engineering curriculum, he most certainly would have had instruction on computer circuits, both analog and digital. There were probably analog computers to use, and possibly they were Philbrick units like the one pictured above, thanks to the connection to the university via Loeb Julie. Where did Moog come up with the idea to make his first synths modular? Did he spend some time with a Philbrick analog computer at Columbia? Did he, perhaps, try to coax sound synthesis out of it?
Buchla is even more of a puzzle. There is almost no information available on the Internet about what he did prior to founding Buchla and Associates in 1962. It is known that he was involved with Morton Subnotick and the San Francisco Tape Music Center, which was a tape studio and had little if anything to do with analog computers. He was involved in some way with the University of California, Berkeley (it's not clear if he was actually a student or faculty there or not), which at the time was the world's foremost center of nuclear physics research, a field in which a considerable number of analog computers were used to model nuclear reactions. Buchla studied physics (along with several other fields) and probably would have come into contact with the nuclear physics program's analog computers. To what extent this influenced his later thinking about synthesizers is difficult to say.
So to answer our question: did analog computers influence the development of analog synths? The answer, at this point, is "maybe". We know that Pearlman was heavily involved in analog computers, but he came in a little after Moog and Buchla. We know that Moog was at Columbia at a time when the school was involved in both analog computing and electronic music, and we can see similarities between his modular synth designs and some of the modular computer designs that he might have worked with. Buchla is less certain, but he probably would have at least seen analog computers at Berkeley.
For more information about George Philbrick and his pioneering company (it's a worthwhile read for anyone interested in electrical engineering history), see the Philbrick Archive at www.philbrickarchive.org.