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33 Coulomb FunctionsNotation

§33.1 Special Notation

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Referenced by:
§33.24, §33.5(iv)
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http://dlmf.nist.gov/33.1

(For other notation see Notation for the Special Functions.)

k,\ell nonnegative integers.
r,x real variables.
\rho nonnegative real variable.
\epsilon,\eta real parameters.
\mathop{\psi\/}\nolimits\!\left(x\right) logarithmic derivative of \mathop{\Gamma\/}\nolimits\!\left(x\right); see §5.2(i).
\mathop{\delta\/}\nolimits\!\left(x\right) Dirac delta; see §1.17.
primes derivatives with respect to the variable.

The main functions treated in this chapter are first the Coulomb radial functions \mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right), \mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right), \mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right) (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions \mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right), \mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right), \mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right), \mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right) (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.

Alternative Notations

  1. Curtis (1964a):

    P_{\ell}(\epsilon,r)=(2\ell+1)!\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r% \right)/2^{{\ell+1}}, Q_{\ell}(\epsilon,r)=-(2\ell+1)!\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r% \right)/(2^{{\ell+1}}A(\epsilon,\ell)).

  2. Greene et al. (1979):

    f^{{(0)}}(\epsilon,\ell;r)=\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right), f(\epsilon,\ell;r)=\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right), g(\epsilon,\ell;r)=\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right).

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