Difference between revisions of "Cosine integral"
From specialfunctionswiki
| Line 4: | Line 4: | ||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | File:Ciplot.png|Graph of $\mathrm{Ci}$. |
File:Domain coloring cosine integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{Ci}$. | File:Domain coloring cosine integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{Ci}$. | ||
</gallery> | </gallery> | ||
Revision as of 21:29, 23 May 2016
The cosine integral is defined by $$\mathrm{Ci}(z) = -\displaystyle\int_z^{\infty} \dfrac{\cos t}{t} dt ; |\mathrm{arg} z|<\pi.$$
Graph of $\mathrm{Ci}$.
Domain coloring of analytic continuation of $\mathrm{Ci}$.
Contents
Relationship to other functions
Theorem
The following formula holds: $$\mathrm{Ei}(ix)=\mathrm{Ci}(x)+i\mathrm{Si}(x),$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\mathrm{Ci}$ denotes the cosine integral, and $\mathrm{Si}$ denotes the sine integral.
Proof
References
Videos
Laplace transform of cosine integral
