**Problem 3: Meromorphic Functions**Show that the functions below are meromorphic; that is, the only singularities in the finite \( z \) plane are poles. Determine the location, order, and strength of the poles.**Functions:**(a) \( \frac{z}{z^4 + 2} \)(b) \( \tan z \)(c) \( \frac{z}{\sin^2 z} \)(d) \( \frac{e^z - 1 - z}{z^4} \)(e) \( \frac{1}{2\pi i} \oint_C \frac{w \, dw}{(w^2 - 2)(w - z)} \)*Note:* \( C \) is the unit circle centered at the origin. First find the function for \( |z| < 1 \), then analytically continue the function to \( |z| \geq 1 \).

Given functions:a) b) c) d) e) To find:The singularities and its order.Verify that the function is meromorphic or not.e) Function when .Concept used:The function is said to be meromorphic if the function has only the singularities as poles of finite order.A point is said to be a pole of order if .Only singularities of a rational function are poles.Cauchy's Integral Formula:If the value , then , where is an analytic function.Cauchy's Residue theorem:Let be analytic inside and on a simple closed contour (positive orientation) except for finite number of isolated singularities (poles) and the poles are not on the contour, then .Residue at the pole of order : a) Find the singularity: The singularity is of poles as the function is of rational function.The values of can be determined by using the De-Movire's theorem.Magnitude of the :Argument of the :According to De-Movire's theorem,.Thus, the function has 4 poles of order 1. The poles are .The poles are located on the circle .As the function has the singularity as only the poles, the function is said to be meromorphic.b) The singularity is of poles as the function is of rational function.Poles of the function exists when is zero.Cosine function is equal to zero when .As the power of cosine as 1 in the denominator, the poles are of order 1.As the cosine value of any function takes the real value at any point, the poles are located on the real axis. As the function has only singularities as poles, the function is said to be meromorphic.c) The singularity is of poles as the function is of rational function.As the denominator is as of order 2, the poles are of order 2As the function has zero at and it is a pole of order 2, the resultant order of the pole is 1.The poles are of order 2.As the sine value of any function takes the real value at any point, the poles are located on the real axis. As the function has only singularities as poles, the function is said to be meromorphic.d) The singularity is of poles as the function is of rational function.The pole of the function is 0 and of order 4.Also, the function has zero at .Thus, the pole is of order 3.As the pole is only a point 0, the pole located at the origin.As the function has only singularities as poles, the function is said to be meromorphic.Given integral is that , where is of unit circle.The function as the integrand is .The poles of the integrand is determined when .i.e., is of order 1 and are of order 1.When , the poles is never present in the contour inside the unit circle.Hence, the function can be re-written as .All the points inside the unit circle are present in the contour of .It is clear that, the numerator of is analytic.When , and are present in the contour.When , the function is of .When , the function can be re-written as .When , the function can be re-written as .Now, the function at is the sum of all values of integral (Residues) by cauchy residue theorem.When , Hence, the required function is .The function has simple poles and it is located on real axis.The function is said to be meromorphic as it has only singularity as poles.a) The poles are and each of order 1.They are located in the circle of radius .It is a meromorphic function.b) The poles are and each of order 1.They are located on real axis.It is a meromorphic function.c) The pole and of order 1. The poles and each of order 1.They are located on real axis.It is a meromorphic function.d) The pole is and of order 3.It is located at origin.It is a meromorphic function.e) The function is that .The poles are and each of order 1.It is located on real axis.It is meromorphic function.