Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Trigonometric integral
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Efficient evaluation == [[Padé approximant]]s of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015),<ref name=RoweEtAl2015>{{cite journal |last1=Rowe |first1=B. |display-authors=etal |title=GALSIM: The modular galaxy image simulation toolkit |journal=Astronomy and Computing |year=2015 |volume=10 |page=121 |doi=10.1016/j.ascom.2015.02.002 |arxiv=1407.7676 |bibcode=2015A&C....10..121R|s2cid=62709903 }}</ref> are accurate to better than {{math|10<sup>−16</sup>}} for {{math|0 ≤ ''x'' ≤ 4}}, <math display="block">\begin{array}{rcl} \operatorname{Si}(x) &\approx & x \cdot \left( \frac{ \begin{array}{l} 1 -4.54393409816329991\cdot 10^{-2} \cdot x^2 + 1.15457225751016682\cdot 10^{-3} \cdot x^4 - 1.41018536821330254\cdot 10^{-5} \cdot x^6 \\ ~~~ + 9.43280809438713025 \cdot 10^{-8} \cdot x^8 - 3.53201978997168357 \cdot 10^{-10} \cdot x^{10} + 7.08240282274875911 \cdot 10^{-13} \cdot x^{12} \\ ~~~ - 6.05338212010422477 \cdot 10^{-16} \cdot x^{14} \end{array} } { \begin{array}{l} 1 + 1.01162145739225565 \cdot 10^{-2} \cdot x^2 + 4.99175116169755106 \cdot 10^{-5} \cdot x^4 + 1.55654986308745614 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 3.28067571055789734 \cdot 10^{-10} \cdot x^8 + 4.5049097575386581 \cdot 10^{-13} \cdot x^{10} + 3.21107051193712168 \cdot 10^{-16} \cdot x^{12} \end{array} } \right)\\ &~&\\ \operatorname{Ci}(x) &\approx & \gamma + \ln(x) +\\ && x^2 \cdot \left( \frac{ \begin{array}{l} -0.25 + 7.51851524438898291 \cdot 10^{-3} \cdot x^2 - 1.27528342240267686 \cdot 10^{-4} \cdot x^4 + 1.05297363846239184 \cdot 10^{-6} \cdot x^6 \\ ~~~ -4.68889508144848019 \cdot 10^{-9} \cdot x^8 + 1.06480802891189243 \cdot 10^{-11} \cdot x^{10} - 9.93728488857585407 \cdot 10^{-15} \cdot x^{12} \\ \end{array} } { \begin{array}{l} 1 + 1.1592605689110735 \cdot 10^{-2} \cdot x^2 + 6.72126800814254432 \cdot 10^{-5} \cdot x^4 + 2.55533277086129636 \cdot 10^{-7} \cdot x^6 \\ ~~~ + 6.97071295760958946 \cdot 10^{-10} \cdot x^8 + 1.38536352772778619 \cdot 10^{-12} \cdot x^{10} + 1.89106054713059759 \cdot 10^{-15} \cdot x^{12} \\ ~~~ + 1.39759616731376855 \cdot 10^{-18} \cdot x^{14} \\ \end{array} } \right) \end{array}</math> The integrals may be evaluated indirectly via [[Auxiliary function|auxiliary functions]] <math>f(x)</math> and <math>g(x)</math>, which are defined by {| |<math display="block">\operatorname{Si}(x)=\frac{\pi}{2}-f(x)\cos(x)-g(x)\sin(x)</math> | |<math display="block">\operatorname{Ci}(x)=f(x)\sin(x)-g(x)\cos(x) </math> |- |colspan="3" align="center"| <small>''or equivalently''</small> |- |<math display="block">f(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \cos(x) + \operatorname{Ci}(x) \sin(x)</math> | |<math display="block">g(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \sin(x) - \operatorname{Ci}(x) \cos(x)</math> |} For <math>x \ge 4</math> the [[Padé approximant|Padé rational functions]] given below approximate <math>f(x)</math> and <math>g(x)</math> with error less than 10<sup>−16</sup>:<ref name=RoweEtAl2015/> <math display="block">\begin{array}{rcl} f(x) &\approx & \dfrac{1}{x} \cdot \left(\frac{ \begin{array}{l} 1 + 7.44437068161936700618 \cdot 10^2 \cdot x^{-2} + 1.96396372895146869801 \cdot 10^5 \cdot x^{-4} + 2.37750310125431834034 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.43073403821274636888 \cdot 10^9 \cdot x^{-8} + 4.33736238870432522765 \cdot 10^{10} \cdot x^{-10} + 6.40533830574022022911 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 4.20968180571076940208 \cdot 10^{12} \cdot x^{-14} + 1.00795182980368574617 \cdot 10^{13} \cdot x^{-16} + 4.94816688199951963482 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 4.94701168645415959931 \cdot 10^{11} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 7.46437068161927678031 \cdot 10^2 \cdot x^{-2} + 1.97865247031583951450 \cdot 10^5 \cdot x^{-4} + 2.41535670165126845144 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 1.47478952192985464958 \cdot 10^9 \cdot x^{-8} + 4.58595115847765779830 \cdot 10^{10} \cdot x^{-10} + 7.08501308149515401563 \cdot 10^{11} \cdot x^{-12} \\ ~~~ + 5.06084464593475076774 \cdot 10^{12} \cdot x^{-14} + 1.43468549171581016479 \cdot 10^{13} \cdot x^{-16} + 1.11535493509914254097 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ & &\\ g(x) &\approx & \dfrac{1}{x^2} \cdot \left(\frac{ \begin{array}{l} 1 + 8.1359520115168615 \cdot 10^2 \cdot x^{-2} + 2.35239181626478200 \cdot 10^5 \cdot x^{-4} +3.12557570795778731 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.06297595146763354 \cdot 10^9 \cdot x^{-8} + 6.83052205423625007 \cdot 10^{10} \cdot x^{-10} + 1.09049528450362786 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 7.57664583257834349 \cdot 10^{12} \cdot x^{-14} + 1.81004487464664575 \cdot 10^{13} \cdot x^{-16} + 6.43291613143049485 \cdot 10^{12} \cdot x^{-18} \\ ~~~ - 1.36517137670871689 \cdot 10^{12} \cdot x^{-20} \end{array} }{ \begin{array}{l} 1 + 8.19595201151451564 \cdot 10^2 \cdot x^{-2} + 2.40036752835578777 \cdot 10^5 \cdot x^{-4} + 3.26026661647090822 \cdot 10^7 \cdot x^{-6} \\ ~~~ + 2.23355543278099360 \cdot 10^9 \cdot x^{-8} + 7.87465017341829930 \cdot 10^{10} \cdot x^{-10} + 1.39866710696414565 \cdot 10^{12} \cdot x^{-12} \\ ~~~ + 1.17164723371736605 \cdot 10^{13} \cdot x^{-14} + 4.01839087307656620 \cdot 10^{13} \cdot x^{-16} + 3.99653257887490811 \cdot 10^{13} \cdot x^{-18} \end{array} } \right) \\ \end{array}</math>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)