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C. Inequalities

Inequality 1

 (C.1)

Proof. According to the Weierstraß factorization theorem, the cosine function can be written as [101]

 (C.2)

Bearing in mind that all factors are in the range for , the inequality follows directly by neglecting all factors with . The first factor is identical to the right-hand side in (C.1).

Inequality 2

 (C.3)

Proof. For both sides are equal. Hence, it is sufficient to show that the function is monotonically increasing. This is the case, if the first derivative is always non-negative, , which is obviously satisfied.

Inequality 3

 (C.4)

Proof. The denominator is positive for . Thus, the inequality is equivalent to the next statement which is proved in the following for all :

 (C.5) (C.6) (C.7)

Using the product representation of the cosine function (C.2):

 (C.8) (C.9)

Since all factors of this product series are in the range , it is sufficient to show that:

 (C.10) (C.11)

Using for all :

 (C.12) (C.13) (C.14)

Using again :

 (C.15) (C.16) (C.17)

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Otmar Ertl: Numerical Methods for Topography Simulation