May 21, 2024

The Derivative Instead of the Integral: A Revolution in Analysis

The Derivative Instead of the Integral: A Revolution in Analysis

\[ f(x) = \sum_{k=0}^\infty a_k x^k = \lim_{\epsilon \to 0} \sum_{k=0}^\infty a_k x^k e^{\epsilon x} \]

This term can be rewritten with the help of derivatives of the exponential function. The goal is to determine the factors xK To replace from the equation. The figure then looks more complicated, but it proves to be very useful for simplifying the integration later. Here the crucial practical property is that the derivative of the exponential function is again the exponential function:

\[ f(x) = \lim_{\epsilon \to 0}\sum_{k=0}^\infty a_k x^k e^{\epsilon x} = \lim_{\epsilon \to 0}\sum_{k=0}^\infty a_k \left(\frac{d}{d \epsilon}\right)^k e^{\epsilon x}\]

With each derivation of the exponential function with respect to one obtains according to the chain rule a x as a factor. You are about to finish. Because as it turns out, you can return the infinite string to the original function F Rewriting. However, the argument changes: the power chain is no longer dependent on x But it is derivative with respect to:

\[ f(x) = \lim_{\epsilon \to 0}\sum_{k=0}^\infty a_k \left(\frac{d}{d \epsilon}\right)^k e^{\epsilon x} = \lim_{\epsilon \to 0} f \left(\frac{d}{d \epsilon}\right) e^{\epsilon x} \]

You have already reached your goal. By substituting this function into an integral, one can simplify to get rid of the integral altogether. Then F It is no longer dependent on the integration variable x Thus it can be derived from the integral. You only need to integrate over an exponential function – the simplest argument imaginable for integration:

\[ \int_a^b f(x) dx = \lim_{\epsilon \to 0} f \left(\frac{d}{d \epsilon}\right) \int_a^b e^{\epsilon x} dx = \lim_{\epsilon \to 0} f \left(\frac{d}{d \epsilon}\right) \frac{e^{\epsilon b}-e^{\epsilon a}}{\epsilon} \]

In this way, the complex part of the integration can be replaced by a derivative. The difficulty of the task now is to know what F(d/dε). This helps here F It can be rewritten as a power series: \(\sum_{k=0}^\infty a_k \left( \frac{d}{d\epsilon} \right)^k\). That is, the terms of higher derivatives are applied to the expression \(\frac{e^{\epsilon b} -e^{\epsilon a}}{\epsilon}\).

For example, you can ask yourself what would happen if F It is an exponential function. In other words: What is the result of the post?d / dε Applies to any differential function J(ε)? As it turns out (see box below), the result is nothing more than a shift in function J: Hd / dεJ(ε) = J(ε + 1).

Since many functions (such as sine and cosine or hyperbolic functions) can be expressed in terms of exponential functions, their integrals can be computed very quickly using the new method – even if many are concatenated together. Where else is partial integral used over and over again, you can now cleverly calculate derivatives instead.