Consider the function f(z) = 1 / (e^(?/z)

Consider the function f(z) = 1 / (e^(?/z) - 1). (i) Find all isolated singularities of this function and indicate them on the complex plane. Determine their nature (removable, poles (of which order?), essential, etc.). (ii) Find residues of f(z) at those singularities. (iii) Show that z = 0 is non-isolated singularity of f(z).

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00:02 Hi, from the question given that, consider the given function f of z is equal to 1 by e to the power of 5 by z minus 1.

00:14 So for part a, we need to find isolated singularity and also we need to plot the isolated singular points on the complex thing.

00:34 So first, for finding isolated singularity, we have to equate, denominated to be equal.

00:41 To 0 so e to the power of 5 by z minus 1 is equal to 0 so e to the power of 5 by z equal to 1 so this is rewritten as e to the power of 2k 5 i therefore this imply 5 by z is equal to 2k 5 i cancel out the common term so we get that is equal to 1 by 2k i where k equal to 1, 2 and so on.

01:20 Therefore these are the isolator singular point.

01:38 Now we need to graph the these isolated singular point on the complex plane.

01:45 This is follows.

01:47 Now this is the complex plane with the isolated singular point.

01:52 Therefore, each singularity.

01:57 Is a pole of order 1.

02:09 Therefore, we have concluded that the isolated singular points are that is equal to 1 by 2k i where k is equal to 1, 2 and so on and the nature of the singularity is whole of order 1.

02:34 Now we move on to the part b.

02:39 So for part b we need to find residual of f of z at these singular point and residual of f where z is equal to 1 by 2k i since z is equal 2k i 1 by 2k i is a singular point...

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