Math 656 HW8 solutions

# Differentiate this expression directly to obtain the

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Differentiate this expression directly to obtain the familiar result 1 2 2 1 1 1 2 1 1 tan log 2 2 2 ( )( ) 1 d d i z i z d i z i z i z dz dz i i z i i z dz i z i i z i z i z z i z   Show that this equals the reciprocal of the derivative of the inverse function ( z =tan w ) 1 2 2 2 2 2 1 1 1 1 tan cos sin 1 tan 1 tan 1 cos d z w d w dz w z w dw w

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x z 2 = r 2 exp( i 2·0 + )= r 2 exp( i 0 + ) y z 2 = r 2 exp( i π ) z 2 = r 2 exp( i π + )= r 2 exp( i · 0 + ) must constrain the angle to [0, 2 π ) z 2 = r 2 exp( i π )= r 2 exp( i · 2 π ) z = r exp( i θ ) 3. Find the branch point(s) and the possible branch cut(s) of the principal branch of 2 ( ) log( ) f z z Principal branch of the log( w ) jumps along the positive real axis of variable w = z 2 Solving for z , we obtain z w   , which gives two branch cuts, corresponding to negative and positive real axis, connecting the branch points at z =0 and z =
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• Spring '11
• Uldon
• Math, Complex number, real axis, positive real axis, negative real axis

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