Liquid Crystal Displays. Ernst Lueder

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or with ξ₁ and ξ_s in Equation (4.18),

(4.29)

      and

(4.30)

with φ from Equation (4.22).

We continue with the normalized matrices M:

(4.31)

Based on Equation (4.17), we can represent T(ε)R(ε) with the known matrices D in Equation (4.26) and M in Equation (4.31) as

      (4.32)

      [T (ε) R (ε)]^2πd/pε in Equation (4.14) assumes the form

      (4.33)

      The evaluation of D^2πd/pε, with dε in Equation (4.5) and D in Equation (4.26), provides

      (4.34)

      K′ in Equation (4.25) leads to

As both the numerator and denominator tend to zero for ε → 0, the application of Hopital’s rule is needed, yielding

(4.35)

      The repetition of Hopital’s rule provides

As the denominator of the last equation is identical with lim_{ε → 0}k′/(2π/p) in Equation (4.35), we obtain

(4.36)

      ensuring

      (4.37)

The limit of M in Equation (4.31) is calculated in the following steps with φ = arctan as the elements mik in the unnormalized matrix M in Equation (4.28) and the magnitudes in Equations (4.29) and (4.30) tend to zero for ε → 0, we choose r₁ = r₂ 1/ε and evaluate the limits of the numerator and denominator in Equation (4.31) separately according to Hopital’s rule. By doing so, we obtain for the numerators

      (4.38)

      and

      With similar calculations as performed for D, we obtain the limit as

      (4.39)

      and in the same way, also

      (4.40)

For the magnitudes the evaluation of the limit value is performed at the square of the magnitudes in Equations (4.29) and (4.30), again leading with similar calculations as for D to

      (4.41)

      and

      (4.42)

      Hence, the normalized M is, for ε → 0,

(4.43)

      from which, as M is a unitary matrix, we obtain

(4.44)

Inserting Equations (4.29), (4.30), (4.43) and (4.44) into Equation (4.28) provides, for ε → 0,

(4.45)

with