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CoulombF

Regular Coulomb wave function

 Calling Sequence CoulombF(L, n, p)

Parameters

 L - algebraic expression n - algebraic expression p - algebraic expression

Description

 • CoulombF is the regular Coulomb wave function. It satisfies the differential equation

$y\text{'}\text{'}\left(x\right)+\left(1-2\frac{n}{x}-\frac{L\left(L+1\right)}{{x}^{2}}\right)y\left(x\right)=0$

Examples

 > $\mathrm{CoulombF}\left(0,0,x\right)$
 ${\mathrm{sin}}{}\left({x}\right)$ (1)
 > $\mathrm{CoulombF}\left(3,0,x\right)$
 $\frac{\sqrt{{\mathrm{\pi }}{}{x}}{}\left({\mathrm{cos}}{}\left({x}\right){}{{x}}^{{3}}{-}{6}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{2}}{-}{15}{}{\mathrm{cos}}{}\left({x}\right){}{x}{+}{15}{}{\mathrm{sin}}{}\left({x}\right)\right)}{\sqrt{{\mathrm{\pi }}}{}{{x}}^{{7}}{{2}}}}$ (2)
 > $\mathrm{CoulombF}\left(2.4,1.7,-3.2+I\right)$
 ${0.001110433908}$ (3)

The derivative of the CoulombF function.

 > $\mathrm{diff}\left(\mathrm{CoulombF}\left(L,n,x\right),x\right)$
 $\frac{\left(\frac{{\left({L}{+}{1}\right)}^{{2}}}{{x}}{+}{n}\right){}{\mathrm{CoulombF}}{}\left({L}{,}{n}{,}{x}\right)}{{L}{+}{1}}{-}\frac{\sqrt{{\left({L}{+}{1}\right)}^{{2}}{+}{{n}}^{{2}}}{}{\mathrm{CoulombF}}{}\left({L}{+}{1}{,}{n}{,}{x}\right)}{{L}{+}{1}}$ (4)
 > $\mathrm{Coulomb_ODE}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(1-\frac{2n}{x}-\frac{L\left(L+1\right)}{{x}^{2}}\right)y\left(x\right)$
 ${\mathrm{Coulomb_ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\left({1}{-}\frac{{2}{}{n}}{{x}}{-}\frac{{L}{}\left({L}{+}{1}\right)}{{{x}}^{{2}}}\right){}{y}{}\left({x}\right)$ (5)
 > $\mathrm{odetest}\left(y\left(x\right)=\mathrm{CoulombF}\left(L,n,x\right),\mathrm{Coulomb_ODE}\right)$
 ${0}$ (6)
 > $\mathrm{dsolve}\left(\mathrm{Coulomb_ODE}\right)$
 ${y}{}\left({x}\right){=}\mathrm{c__1}{}{\mathrm{WhittakerM}}{}\left({I}{}{n}{,}{L}{+}\frac{{1}}{{2}}{,}{2}{}{I}{}{x}\right){+}\mathrm{c__2}{}{\mathrm{WhittakerW}}{}\left({I}{}{n}{,}{L}{+}\frac{{1}}{{2}}{,}{2}{}{I}{}{x}\right)$ (7)
 > $\mathrm{CoulombF}\left(L,n,x\right)=\mathrm{convert}\left(\mathrm{CoulombF}\left(L,n,x\right),\mathrm{WhittakerM}\right)$
 ${\mathrm{CoulombF}}{}\left({L}{,}{n}{,}{x}\right){=}\frac{{{2}}^{{L}}{}\left|{\mathrm{\Gamma }}{}\left({1}{+}{L}{+}{I}{}{n}\right)\right|{}{{x}}^{{L}{+}{1}}{}{\mathrm{WhittakerM}}{}\left({I}{}{n}{,}{L}{+}\frac{{1}}{{2}}{,}{2}{}{I}{}{x}\right)}{{{ⅇ}}^{\frac{{n}{}{\mathrm{\pi }}}{{2}}}{}{\mathrm{\Gamma }}{}\left({2}{+}{2}{}{L}\right){}{\left({2}{}{I}{}{x}\right)}^{{L}{+}{1}}}$ (8)

References

 Abramowitz, M., and Stegun, I. eds. Handbook of Mathematical Functions. New York: Dover publications.

 See Also