How to Solve and Integrate the Exponential Fourier Series

Understanding and solving the integral of exponential Fourier series can be complex, especially when dealing with functions of the form ( frac{e^{-x/a}}{1 - ax} ). This article provides a step-by-step guide to integrating such expressions, using advanced mathematical techniques and functions, including the Exponential Integral.

Step-by-Step Solution

The given integral is ( int_0^b frac{e^{-x/a}}{1 - ax} , dx ). The solution involves several transformations and the introduction of the Exponential Integral function. Here's how the process works:

Step 1: Variable Substitution

First, let ( t 1 - ax ). This transforms the integral and introduces a new variable ( t ) that simplifies the expression. The differential ( dt ) is found to be ( dt -a , dx ), which changes the limits of integration and the integrand. With these transformations, the integral becomes:

[ int_0^b frac{e^{-x/a}}{1 - ax} , dx int_1^{1-ab} frac{e^{-t-1/a^2}}{t} , frac{dt}{a} frac{e^{1/a^2}}{a} int_1^{1-ab} frac{e^{-t/a^2}}{t} , dt. ]

Step 2: Further Substitution

To simplify the integrand further, use the substitution ( w frac{t}{a^2} ). The differential ( dw ) is calculated as ( dw frac{1}{a^2} , dt ), which leads to the integral being rewritten in terms of ( w ):

[ frac{e^{1/a^2}}{a} int_{frac{1}{a^2}}^{frac{1-ab}{a^2}} frac{e^{-w}}{w} , dw. ]

Step 3: Utilize the Exponential Integral Function

The integral ( int frac{e^{-w}}{w} , dw ) is known as the Exponential Integral function, denoted as ( text{Ei}(x) ). Therefore, the integral can be expressed as:

[ frac{e^{1/a^2}}{a} Big[int_{frac{1}{a^2}}^{infty} frac{e^{-w}}{w} , dw - int_{frac{1-ab}{a^2}}^{infty} frac{e^{-w}}{w} , dw Big]. ]

Final Result

By applying the definition of the Exponential Integral function, the solution to the original integral can be written as:

[ int_0^b frac{e^{-x/a}}{1 - ax} , dx frac{e^{1/a^2}}{a} Big[ text{Ei}Big( -frac{1-ab}{a^2} Big) - text{Ei}Big( -frac{1}{a^2} Big) Big]. ]

Conclusion

The solution provided demonstrates the use of substitution techniques and the Exponential Integral function to solve a complex integral. Understanding these advanced mathematical tools is essential for tackling problems in various fields, including physics and engineering, where similar forms of integrals often appear.