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f(z) is a function of complex variable z given by
- a)z = –1 is a simple pole
- b)z = –1 is a branch point
- c)z = 1 is a branch point
- d)z = 1 is analytic in the region
Correct answer is option 'C'. Can you explain this answer?
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f(z) is a function of complex variable z given bya)z = –1 is a s...
a) z = -1 as a simple pole: The denominator of the function includes the factor (z + 1)². A simple pole has first order in the denominator, but here at z=-1 we have a factor of second order, which means z=-1 is actually a pole of order 2, not a simple pole. So, this statement is false.
b) z = -1 as a branch point: The term √(z-1) introduces a branch point at z=1; This is simply the point at which the function is not analytic due to the function's non-integer power. However, z=-1 does not cause the function to be non-integer and hence is not a branch point. So, this statement is also false.
c) z = 1 as a branch point: As mentioned above, the term √(z-1) introduces a branch point at z=1 due to the square root function (or, more generally given that we're in the complex plane, because of the multi-valued nature of the complex power function). So z=1 is indeed a branch point. Therefore, this statement is true.
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f(z) is a function of complex variable z given bya)z = –1 is a simple poleb)z = –1 is a branch pointc)z = 1 is a branch pointd)z = 1 is analytic in the regionCorrect answer is option 'C'. Can you explain this answer?
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