Difference between revisions of "Exponential integral E"
From specialfunctionswiki
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$$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi$$ | $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi$$ | ||
and | and | ||
| − | The exponential integral is related to the [[logarithmic integral]] by the formula | + | $$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z<\pi|.$$ |
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The exponential integral $\mathrm{Ei}$ is related to the [[logarithmic integral]] by the formula | ||
$$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$ | $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=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 /> | ||
Revision as of 01:32, 2 February 2015
The exponential integrals are $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi$$ and $$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z<\pi|.$$
Properties
Proposition: The exponential integral $\mathrm{Ei}$ is related to the logarithmic integral by the formula $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$
Proof: █
