Plasma Dispersion Function

(8.39) |

which is defined as it is written for , and is analytically continued for . This function is known as the

In view of the importance of the plasma dispersion function, and its regular appearance in the literature of plasma physics, it is convenient to briefly examine its main properties. We, first of all, note that if we differentiate with respect to then we obtain

(8.40) |

which yields, on integration by parts,

If we let tend to zero from the upper half of the complex plane, then we get

Of course, the principal part integral is zero because its integrand is an odd function of .

Integrating the linear differential equation (8.41), which possesses an integrating factor , and using the boundary condition (8.42), we obtain an alternative expression for the plasma dispersion function:

(8.43) |

Making the substitution in the integral, and noting that

(8.44) |

we finally arrive at the expression

(8.45) |

This formula, which relates the plasma dispersion function to an error function of imaginary argument (Abramowitz and Stegun 1965b), is valid for all values of .

For small , we have the expansion (Huba 2000c)

(8.46) |

For large , where , the asymptotic expansion for is written (Huba 2000c)

Here,

(8.48) |

In deriving our previous expression (8.32) for the Landau damping rate, we, in effect, used the first few terms of the asymptotic expansion (8.47).

The properties of the plasma dispersion function are specified in exhaustive detail in a well-known book by Fried and Conte (Fried and Conte 1961).