samples/texts/2974107/page_11.md

We use the relations :

$$I_0=\frac{1}{(2T_i{)}^{1\mathrm{/}2}}Z\left(\frac{u}{(2T_i{)}^{1\mathrm{/}2}}\right),I_1=1+u{I}_0,I_2=u(1+u{I}_0),I\_0\; =\; \backslash frac\{1\}\{(2T\_i)^\{1/2\}\}\; Z\backslash left(\backslash frac\{u\}\{(2T\_i)^\{1/2\}\}\backslash right),\; \backslash quad\; I\_1\; =\; 1\; +\; uI\_0,\; \backslash quad\; I\_2\; =\; u(1+uI\_0),$$

with

$$Z(u)=\frac{1}{\sqrt{\pi }}\int \frac{\exp (-x^2)}{x-u}dx=i\sqrt{\pi }\exp (-u^2)(1-\mathrm{erf}(-iu)),Z(u)\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\; \backslash frac\{\backslash exp(-x^2)\}\{x-u\}\; dx\; =\; i\backslash sqrt\{\backslash pi\}\; \backslash exp(-u^2)(1\; -\; \backslash operatorname\{erf\}(-iu)),$$

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x\exp (-t^2)dt\mathrm{.}\backslash operatorname\{erf\}(x)\; =\; \backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\_\{0\}^\{x\}\; \backslash exp(-t^2)\; dt.$$

The dispersion relation is, putting $\phi = \phi_{m,n,\omega}$, for convenience,

$$A=-{\mathrm{\partial }}_r^2\varphi -(\frac{1}{r}+\frac{{\mathrm{\partial }}_r{n}_0}{n_0}){\mathrm{\partial }}_r\varphi +\frac{m^2}{r^2}\varphi +\frac{1}{T_e}\varphi =[-\frac{1}{T_i}(1+zZ(z))+\frac{m}{k^{\ast }r{B}_0}(Z(z)(\frac{{\mathrm{\partial }}_r{n}_0}{n_0}-\frac{{\mathrm{\partial }}_r{T}_i}{2T_i})+z(1+zZ(z))\frac{{\mathrm{\partial }}_r{T}_i}{T_i})]\varphi ,A\; =\; -\backslash partial\_r^2\; \backslash phi\; -\; \backslash left(\backslash frac\{1\}\{r\}\; +\; \backslash frac\{\backslash partial\_r\; n\_0\}\{n\_0\}\backslash right)\; \backslash partial\_r\; \backslash phi\; +\; \backslash frac\{m^2\}\{r^2\}\; \backslash phi\; +\; \backslash frac\{1\}\{T\_e\}\; \backslash phi\; \backslash \backslash \; =\; \backslash left[\; -\backslash frac\{1\}\{T\_i\}\; (1+zZ(z))\; +\; \backslash frac\{m\}\{k^*\; r\; B\_0\}\; \backslash left(\; Z(z)\; \backslash left(\; \backslash frac\{\backslash partial\_r\; n\_0\}\{n\_0\}\; -\; \backslash frac\{\backslash partial\_r\; T\_i\}\{2T\_i\}\; \backslash right)\; +\; z(1+zZ(z))\; \backslash frac\{\backslash partial\_r\; T\_i\}\{T\_i\}\; \backslash right)\; \backslash right]\; \backslash phi,$$

with $k^* = (2T_i)^{1/2}k_|$ and $z = \frac{\omega}{k^*}$, and recalling that $k_| = (b_\theta \frac{m}{r} + b_z k)$. Note that the dispersion relation depends on $m$ and $k_|$ and not directly on $n$. This means that taking different values of $\iota$ and $n$ but with same $m$ and $k_|$ will lead to the same dispersion relation.

References

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