Generally speaking, if the antiderivative is easy to find, the Newton-Leibniz formula is the universal method for definite integrals. However, the Newton-Leibniz formula is only suitable for the integration of continuous functions; if the integration interval contains singularities, it no longer holds. For example, let us consider the integral: \int_{-1}^1 \frac{1}{x^2}dx Of course, from a strict mathematical standpoint, this notation is invalid because the integrand is undefined at the origin. However, from a physical perspective, due to symmetry, we are certain that: \int_{-1}^1 \frac{1}{x^2}dx = 2\int_{0}^1 \frac{1}{x^2}dx = \lim_{\varepsilon\to 0} 2\int_{\varepsilon}^1 \frac{1}{x^2}dx This leads to the conclusion that the integral diverges. While this treatment is acceptable to some extent, it is not entirely satisfactory because it necessitates piecewise division. Is there a way to handle this situation directly? Indeed, there is. By introducing a parameter and eventually letting the parameter approach zero, we consider the integral with a parameter: \int_{-1}^1 \frac{1}{x^2+\varepsilon^2}dx As long as the parameter is positive, this integrand is continuous everywhere on \mathbb{R}, meaning the singularity has vanished. In this way, the Newton-Leibniz formula becomes applicable again: \int_{-1}^1 \frac{1}{x^2+\varepsilon^2}dx = \left.\frac{1}{\varepsilon}\arctan\left(\frac{x}{\varepsilon}\right)\right|_{-1}^{1} Considering the case where \varepsilon\to 0, we automatically obtain the conclusion that the integral diverges.
However, when considering the integral: \int_{-1}^1 \frac{1}{x}dx The aforementioned trick no longer works. This is because after adding a parameter: \int_{-1}^1 \frac{1}{x+\varepsilon}dx The singularity still exists within the integration region; it has merely been shifted slightly. How can this problem be solved? Physicists devised a very ingenious method (I am unsure whether it was a mathematician or a physicist who first conceived it, but I encountered it in a physics tutorial): Flip to a new dimension! The term they add is: \int_{-1}^1 \frac{1}{x+i\varepsilon}dx This places the integrand into the complex plane! This function is continuous everywhere on the real axis, and thus the Newton-Leibniz formula takes effect once more: \int_{-1}^1 \frac{1}{x+i\varepsilon}dx = \left.(\ln |x+i\varepsilon|)\right|_{-1}^1 = 0
The advantage of this approach is that, in mathematical processing, it makes the singularity disappear, allowing various infinities to cancel out naturally, leaving a net result.
This treatment is indeed quite non-rigorous. However, in physics, especially in Quantum Field Theory, when faced with various infinities, even basic calculation becomes a problem—how can one afford to worry about rigor? Physicists must always calculate the answer first, see if it matches reality, and only then consider issues such as rigor and axiomatization. Therefore, as a product of a “transition period,” this calculation technique is very necessary.
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