Fourier transform is a widely used tool but there is no consensus about the signs in the exponential, neither about the pre-factors in front of the integral. You see here my choice for the Fourier transform, f tilde of capital omega of the function f of tau. Time tau and angular frequency omega are conjugate variables. I have a plus sign in the exponential and one over root 2 pi as a prefactor. Once one has made the choice for the direct transform, the form of the inverse transform is imposed. With my choice, the prefactor in the reciprocal, which is another name for the inverse, is the same, and there is a minus sign in the exponential. I use these forms consistently in this course and my textbooks and lecture notes as well. Other authors may use the opposite signs in the exponentials and different prefactors. Similar forms exist for the spatial Fourier transform where conjugate variables are position r and k-vector k. They correspond to the Fourier expansions we use in this course. The fact that we can either calculate a Fourier transform or its reciprocal is very useful because it often happens that the calculation is much easier one way, then the other way. So when you need to know the exact value of a Fourier transform, you better choose the simplest of the two calculations, and trust our colleagues mathematicians, that the reciprocal formula holds, under the proper conditions of course. Let us take the example of a decaying exponential which we have just encountered. Remember that H of tau is 1 for positive tau and null for negative tau, so the Fourier transform writes as the integral of an exponential from 0 to infinity. The result of the integration is obtained immediately as 1 over the coefficient in the exponent. You may feel uncomfortable with the complex value of the exponent, but as long as the integration variable is real, you can use standard calculus results. Finally we have the expression of that function and its Fourier transform. I have multiplied the numerator and denominator by -i to obtain a more usual form. You have just seen that it takes no effort to find the Fourier transform of a decaying exponential written here. If you had tried to calculate the other way, it would have been much more difficult. In fact it demands to master integration in the complex plane, a not too elementary mathematical tool. Fortunately, if we have done the calculation one way, we can trust the reciprocal. And we thus know the result of that difficult integral without calculating it. It is H of tau times the decaying exponential, with inverse time constant gamma times a prefactor minus i. By the way, do you know how Fourier invented the transform that has his name? In 1802, Napoleon had appointed him a Préfet kind of a governor of the county of Grenoble in the French Alps. A polymath, educated at Ecole Polytechnique, Fourier had always been very active in mathematics and physics, and in addition to being an administrator, he found time to do experiments on the propagation of heat. When he wrote an equation to describe that propagation, he found that it was a partial derivative equation, problem without known solution. In order to find solutions, he invented the expansion that we still use. His book the Théorie Analytique De La Chaleur is considered a masterpiece of the history of science. It was published in 1822 after he became the provost of the University of Grenoble which he had founded. This university is nowadays named Université Joseph Fourier. An English translation of his 1822 book was published in 1878 and it is still available. I have read the original version of this book in French. If you already know Fourier theories and Fourier transform, it is a wonder to discover how Fourier proceeds to develop his method in order to solve heat propagation problems.