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ANALYTIC FUNCTIONS 323 periodic function z = z (s) of a parameter s. Except at a finite number of corners this function shall have a continuous derivative T (s) ' which, at each corner, is continuous on both sides. We can assume dz z = ds We set Az (s) = Then we have (2. 3) 'KURT FRIEDRICHS [May z (s + a) -z (s) Az (s) = (s + a) -z (s) Az (s) -z (s) \ g-, 4 3. 5 -^ Az (s) ^-4 4 We choose a number p >0 in such a way that | s' -s | ^ o-as | z (s') -z (s) | g p and a number T > 0 in such a way that and (2. 4) T Consequently we have (2. 4) 'Az (s') -Az (s) rs7> iS -as 40 ^ <r. -|s'-s|+^-|5'-s (+ -'-t\ 4 ' ' 40 ' 4 ' ' ce e -+ -+ -4 40 4 (' -) + * (/'-0|. Now let z' = z. Then we have | z (s') -z (s) I = I t'Az (s') -/Az (s) | ^ /' | Az (s') | + /1 Az (j) | 5 ^ 27-^ p. 4
Kurt Friedrichs (Sat,) studied this question.