       Problems Using Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg117251] Problems Using Mathematica
• From: Crystal <sbi.afolabi at gmail.com>
• Date: Sat, 12 Mar 2011 05:34:11 -0500 (EST)

```Hello Friends,
I'm almost racking my head against the wall trying to solve this
problem... maybe because I'm new to Mathematica but there's no point
trying to learn without help from forums like this.

Here are my problems (I'm hoping the questions should render well in
Mathematica environment with copy/paste):

Problem 1

\[Bullet] Consider the following set of equations

{A E^(\[Alpha] Subscript[L, 1]) - C Cos[\[Gamma] Subscript[L, 1]] -
B Sin[\[Gamma] Subscript[L, 1]] == 0,
A E^(\[Alpha] Subscript[L, 1]) \[Alpha] -
B \[Gamma] Cos[\[Gamma] Subscript[L, 1]] +
C \[Gamma] Sin[\[Gamma] Subscript[L, 1]] ==
0, -D E^(-\[Alpha] Subscript[L, 2]) + C Cos[\[Gamma] Subscript[L,
2]] +
B Sin[\[Gamma] Subscript[L, 2]] == 0,
D E^(-\[Alpha] Subscript[L, 2]) \[Alpha] +
B \[Gamma] Cos[\[Gamma] Subscript[L, 2]] -
C \[Gamma] Sin[\[Gamma] Subscript[L, 2]] == 0}

where

\[Alpha]^2 + \[Gamma]^2 == v^2

with given v and

L == -Subscript[L, 1] + Subscript[L, 2]

Derive the eigenvalue equation in terms of \[Gamma] and L with v as
the parameter.

Problem 2

\[Bullet] Given the reflection amplitude given by

B[z] == -((I E^(-(1/2) I z \[CapitalDelta]\[Beta]) A Sinh[s (L -
z)]
\!\(\*SuperscriptBox["\[Kappa]", "*"]\))/(
s Cosh[L s] + 1/2 I (I g + \[CapitalDelta]\[Beta]) Sinh[L s]))

where

s == Sqrt[-(1/
4) (I g + \[CapitalDelta]\[Beta])^2 + \[LeftBracketingBar]\[Kappa]
\
\[RightBracketingBar]^2]

the reflection gain is given by

Subscript[G, refl] == \[LeftBracketingBar]B/A\
[RightBracketingBar]^2

For simplicity, put \[Kappa]=1,L=1 and find the lowest three numerical
solutions of \[CapitalDelta]\[Beta] and g for infinite gain.

Thanks so much

```

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