Difference between revisions of "Cosine integral"
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The cosine integral is defined by | The cosine integral is defined by | ||
| − | $$\mathrm{Ci}(z) = -\displaystyle\int_z^{\infty} \dfrac{\cos t}{t} | + | $$\mathrm{Ci}(z) = -\displaystyle\int_z^{\infty} \dfrac{\cos t}{t} \mathrm{d}t, \quad |\mathrm{arg} z|<\pi.$$ |
<div align="center"> | <div align="center"> | ||
Revision as of 21:29, 23 May 2016
The cosine integral is defined by $$\mathrm{Ci}(z) = -\displaystyle\int_z^{\infty} \dfrac{\cos t}{t} \mathrm{d}t, \quad |\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
