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  3. 1 2 3 pl k lll find the zeros and orders of zeros

1 2 3 PL K LLL Find the zeros and orders of zeros of the...

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1 2 3 PL K LLL Find the zeros and orders of zeros of the following functions a g e 1 h b c d e 2 2 sinz f cos z 1 Cos 2 1 2 COS sinh2z cosh z Log z 2 d Note that g 2 COS 2 1 0 iff z 2k for k Z We will show that all the zeros are double zeros First observe that g 2 so g 2km sin 2k 0 for all ke Z But g 2 cos z and so g 2km cos 2k 1 for all ke Z Hence all zeros are double zeros principal value g e 1 2 thus a simple zero at 0 Since e 1 is periodic we have simple zeros at i2 k k E Z i Logz 0 when z 1 and Arg z 0 thus at z 1 We write Log z Log 1 1 2 1 z 2 1 3 so we have a simple

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