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# C. Inequalities

Inequality 1 (C.1)

Proof. According to the Weierstraß factorization theorem, the cosine function can be written as (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)     Next: Bibliography Up: Dissertation Otmar Ertl Previous: B.2 Root Finding

Otmar Ertl: Numerical Methods for Topography Simulation