Difference between revisions of "Exponential integral E"
From specialfunctionswiki
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The exponential integrals are | The exponential integrals are | ||
| − | $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi$$ | + | $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi,$$ |
| + | $$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z<\pi|,$$ | ||
and | and | ||
| − | $$ | + | $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} dt.$$ |
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
<strong>Proposition:</strong> The exponential integral $\mathrm{Ei}$ is related to the [[logarithmic integral]] by the formula | <strong>Proposition:</strong> The exponential integral $\mathrm{Ei}$ is related to the [[logarithmic integral]] by the formula | ||
| − | $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)) | + | $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)); n=0,1,2,\ldots, \mathrm{Re}(z)>0$$ |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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=Videos= | =Videos= | ||
[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br /> | [https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br /> | ||
| + | |||
| + | =References= | ||
| + | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions] | ||
Revision as of 01:35, 2 February 2015
The exponential integrals are $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi,$$ $$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z<\pi|,$$ and $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} dt.$$
Properties
Proposition: The exponential integral $\mathrm{Ei}$ is related to the logarithmic integral by the formula $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)); n=0,1,2,\ldots, \mathrm{Re}(z)>0$$
Proof: █
Videos
Laplace transform of exponential integral
