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Complex analysis is a branch of mathematics that studies functions of complex numbers. It is sometimes known as the theory of functions of a complex variable. Many branches of mathematics benefit from it. Complex analysis is concerned with analytic and harmonic functions of a complex variable in particular.
If a complex function is complex differentiable at every point in R, it is said to be analytic on that region. The words “analytic function” and “holomorphic function,” “differentiable function,” and “complex differentiable function” are occasionally used interchangeably. While many mathematicians prefer Harmonic Function to analytic function, analytic appears to be widely used by physicists and engineers.
We will learn about complex functions, analytic functions, and harmonic functions in this article.
Complex Functions
A complex function is one that goes from one complex number to another.
The domain values z and their range pictures f(z) can be divided into real and imaginary parts for any complex function:
z = x + iy
f(z) = f(x + iy) = u(x,y) + iy(x,y)
x, y, u(x, y), v(x, y) are all real-valued variables.
Complex Integration
Complex functions include complex integration. The obvious extension of actual integration is complex integration. Because a complex number represents a point on a plane and a real number represents a value on the real line, the complex domain equivalent of a single real integral is always a path integral.
For a≤t≤b, consider a contour C parametrized by z(t) = x(t) + iy(t). The complex number is defined as the integral of the complex function along C.
∫C fzdz = ∫ba f(z(t)z'(t)dt
Analytic Functions
An analytic function is a function that is locally provided by a convergent power series in mathematics. Real analytic functions and complex analytic functions both exist. Although all functions are endlessly differentiable, complex analytic functions have features that are not shared by real analytic functions. For every x0 in its domain.
In the complex plane, f(z) is said to be analytic in a region R if it has a derivative at each point of R and is single valued. The function f(z) is said to be analytic at that point if z is an interior point of a region where f(z) is analytic.
A function f is real analytic on an open set D in the real line if it can be written for a)ny x0∈D.
f(x) = ∑∞n=0 an(x – x0)n = a0 + a1(x – x0) + a2(x – x0)2 + a3(x – x0)3+..
The coefficients a0,a1,… are real values, and the series converges to f(x) for x in the vicinity of x0.
A real analytic function, on the other hand, is an endlessly differentiable function that has the Taylor series at any point x0 in its domain.
T(x) = ∑∞n=0 [f(n)x0/n!] (x – x0)n
A function f defined on a subset of the real line is said to be real analytic at a point x if f is real analytic in the neighbourhood D of x.
Analytic functions’ properties
Analytic functions’ sums, products, and compositions are all analytic.
The inverse of an invertible analytic function whose derivative is nowhere zero is analytic, as is the reciprocal of an analytic function that is nowhere zero.
Any smooth analytic function is endlessly differentiable. The reverse is not true for real functions; in fact, when compared to all real infinitely differentiable functions, real analytic functions are sparse.
Harmonic function
A harmonic function is a two-variable mathematical function its value at any position equals the average of its values along any circle centred on that point, assuming the function is defined within the circle.
Because this average involves an infinite number of points, it must be calculated using an integral, which represents an infinite sum
If and only if, a complex function f(u) is called harmonic, it must satisfy the following equation:
(∂2u/∂x2) + (∂2u/∂y2) = 0
Every harmonic function is the real part of a holomorphic function in a simply linked domain in the setting of holomorphic functions.
Harmonic function properties
The following are some key properties of harmonic functions:
All partial derivatives of f are harmonic functions on U if f is a harmonic function on U.
In open sets, harmonic functions are infinitely differentiable. Harmonic functions are, in fact, true analytic functions.
Conclusion
In this article we learned that, Complex analysis is a branch of mathematics that studies functions of complex numbers. It is sometimes known as the theory of functions of a complex variable. Complex analysis is concerned with analytic functions of a complex variable and Harmonic functions since a differentiable function of a complex variable is equivalent to its Taylor series. In mathematics, analytic functions are crucial for solving two-dimensional problems.
