ε = 1 - w pe 2 /k 2 [ò du {du f o d(u) / (u - w/ k )]
A space physicist wants to show the di-electric function can be expressed in real and imaginary parts such that: ε r + i ε i = 0
Show this using integration by parts of the equation for ε and then do a Taylor expansion.
Solution: 0 =
1 - w e 2 /k 2 P ò¥-¥ du {du f o d(u) / (u - w/ k ) -πi (w e 2 /k 2) du f o ] u= w r / k
1 - w e 2 /k 2 ò du f o (u) / (1 - ur / w ) 2 - πi (w e 2 /k 2) du f o ] u= w r / k = 0
1 - w e 2 /w r 2 (1 + 3 k 2 v e 2 /w e 2 ) - πi (w e 2 /k 2) d f o / du ] u= w r / k = 0
And:
i ε i = π i (w e 2 /k 2) d f o / du
If | w r | < < ε r
We can do a Taylor expansion about w r :
->
e r (k, w r ) + (w - w r ) ¶ e r / ¶w + i e i + i(w - w r ) ¶ e i / ¶w = 0
Whence: i(w - w r ) ¶ e i / ¶w -> 0
Solving non-zero part:
e r + iw ¶ e r / ¶w] w r + i e i = 0
e r + iw ¶ e r / ¶w] w r + i e i = 0
e r + i e i = 0
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