The residue theorem in complex analysis has a vast application in particular for evaluating the integrals including many integrals from the ordinary calculus. This unit demonstrates this application with a series of examples.

Harmonic functions are tremendously important for the role they play in physics and engineering where they turn out naturally in a variety of context including electronics fluid dynamics and heat transfer. This unit deals with harmonic functions and mean value property for harmonic functions.

Sequence and series representation of a fun ct ion play very important role in many branches of Mathematics. In this unit, we discuss how an analytic function in an open disk can be represented as a Taylor series.

In this unit, we discuss the infinite product and their relations with infinite series. Further, we study Weierstrass Factorization theorem which deals with representation of meromorphic function as an infinite product. We also study canonical product.

In this unit, we discuss the infinite product and their relations with infinite series. Further, we study Weierstrass factorization theorem which
deals with representation of meromorphic function as an infinite product. We also study Canonical product.

The function s such as polynomials, exponential functions, trigonometric functions, inverse trigonometric functions, hyperbolic functions, inverse hyperbolic functions and logarithmic functions are called elementary functions. The functions which can be expressed as the combinations of elementary functions are also called as elementary functions. The
functions which cannot be expressed as element ......

Jensen's formula relates the modulus of f(z) on the circle to the modulus of the zeroes. It has many important applications in the theory of entire functions.

In the previous blocks, we have considered the representation of entire functions as infinite products. In this unit we study the connection between product representation and the rate of growth of the function.

Riemann-zeta function was defined by Riemann in 1859 and he established a relation between its zeros and the distribution of prime numbers in his famous paper entitled 'On the number of primes less than given magnitude'. The values of Riemann zeta function at even positive integers were completed by Euler.

In his paper entitled ' On number of primes less than the given magnitude', Riemann deduced certain functional equations satisfied by Riemann-zeta functions which involves Gamma function. He also relates Riemann-zeta function with Dedekind eta functions. In this unit, we study these types of functional relations.

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