stpaul2coderdojo/feynmans-toolkit-of-tricks

Integration under the differentiation sign is a powerful technique for evaluating integrals that involve a parameter. This technique was used extensively by the Nobel Prize-winning physicist Richard Feynman to solve a wide range of problems in quantum mechanics.

The idea behind integration under the differentiation sign is to differentiate both sides of an integral with respect to a parameter, and then use the resulting differential equation to simplify the integral. This technique can be particularly useful when the integral is difficult to evaluate directly, but the differential equation obtained from differentiating it is easier to solve.

One classic example of Feynman's use of this technique involved the evaluation of a certain class of integrals that arise in quantum electrodynamics (QED). These integrals involve a parameter known as the coupling constant, which determines the strength of the interaction between electrons and photons.

To evaluate these integrals, Feynman differentiated them with respect to the coupling constant, and then used the resulting differential equation to simplify the integral. By repeating this process several times, he was able to obtain a series of equations that allowed him to calculate the value of the integral to high precision.

Another example of Feynman's use of integration under the differentiation sign involved the evaluation of an integral that arose in the study of the Lamb shift, a subtle quantum mechanical effect that causes the energy levels of hydrogen atoms to shift slightly.

To evaluate this integral, Feynman again differentiated it with respect to a parameter, and then used the resulting differential equation to simplify the integral. By applying this technique several times, he was able to obtain a series of equations that allowed him to calculate the value of the integral to high precision.

Feynman's use of integration under the differentiation sign highlights the power and versatility of this technique for evaluating difficult integrals. By differentiating an integral with respect to a parameter, we can often obtain a simpler differential equation that allows us to solve the integral more easily. This technique has applications not only in physics, but in many other areas of mathematics and science as well.