- #1
Euge
Gold Member
MHB
POTW Director
- 2,052
- 207
Evaluate, with proof, the definite integral $$\int_{-\infty}^\infty \frac{e^{ax}}{1 + e^x}\, dx$$ where ##0 < a < 1##.
julian said:The correct answer to the integral
$$
\int_{-\infty}^\infty \frac{e^{a x}}{1 + e^x} dx
$$
can be obtained by closing the contour by a large semi circle in the upper or lower HP. But you need to give an argument for why the semi circle part of the integral can be taken to be zero. I'm not sure how rigorous an argument you can give.
Do you have to be concerned about ##\theta = \pi/2##?pasmith said:The idea would be that [tex]
\begin{split}
\left| \int_0^\pi \frac{e^{aR\cos \theta+iaR\sin\theta}}{1 + e^{R\cos\theta + iR\sin\theta}}iRe^{i\theta}\,d\theta \right|
&\leq \pi R \sup_{\theta \in [0,\pi]} \left| \frac{e^{-(1-a)R\cos \theta}}{e^{-R\cos\theta} + e^{iR\sin\theta}} \right|
\end{split}
[/tex] tends to zero as [itex]R \to \infty[/itex].
julian said:Do you have to be concerned about ##\theta = \pi/2##?
Exponential-type integrals are mathematical expressions that involve an exponential function raised to a power. They are commonly used in many areas of science and engineering, such as in the study of growth and decay processes.
Exponential-type integrals are different from other types of integrals because they involve an exponential function, which can make the integration process more challenging. They also have unique properties and techniques for solving them.
Exponential-type integrals have many applications in science and engineering, such as in the study of radioactive decay, population growth, and signal processing. They are also used in fields like physics, chemistry, and economics.
Exponential-type integrals are used in real-world problems to model and predict various phenomena, such as the growth or decay of populations, the spread of diseases, and the behavior of electrical circuits. They are also used in statistical analysis and data fitting.
There are several techniques for solving exponential-type integrals, including substitution, integration by parts, and the use of special functions such as the gamma function. These techniques can be combined and applied depending on the specific form of the integral.