Complex analysis is known as one of the classical branches of mathematics and analyses complex numbers concurrently with their functions, limits, derivatives, manipulation, and other mathematical properties. Complex analysis is a potent tool with an abruptly immense number of practical applications to solve physical problems. Let’s understand various components of complex analysis one by one here.
Complex numbers
A number of the form x + iy where x, y are real numbers and i2 = -1 is called a complex number.
In other words, z = x + iy is the complex number such that the real part of z is x and is denoted by Re(z), whereas the imaginary part of z is iy and is denoted by I(z).
Modulus and Argument of a Complex Number
The modulus of a complex number z = x + iy is the real number √(x2 + y2) and is denoted by |z|.
The amplitude or argument of a complex number z = x + iy is given by:
arg(z) = θ = tan-1(y/x), where x, y ≠ 0.
Also, the arg(z) is called the principal argument when it satisfies the inequality -π < θ ≤ π, and it is denoted by Arg(z).
Click here to learn about the argument of complex numbers.
Complex Functions
In complex analysis, a complex function is a function defined from complex numbers to complex numbers. Alternatively, it is a function that includes a subset of the complex numbers as a domain and the complex numbers as a codomain. Mathematically, we can represent the definition of complex functions as given below:
A function f : C → C is called a complex function that can be written as
w = f(z), where z ∈ C and w ∈ Z.
Also, z = x + iy and w = u + iv such that u = u(x, y) and v = v(x, y). That means u and v are functions of x and y.
Limits of Complex Functions
Let w = f(z) be any function of z defined in a bounded closed domain D. Then the limit of f(z) as z approaches z0 is denoted by “l”, and is written as
Continuity of Complex Functions
Let’s understand what is the continuity of complex functions in complex analysis.
A complex function w = f(z) defined in the bounded closed domain D, is said to be continuous at a point Z0, if f(z0) is defined
Complex Differentiation
Some of the standard results of complex differentiation are listed below:
- dc/dz = 0; where c is a complex constant
- d/dz (f ± g) = (df/dz) ± (dg/dz)
- d/dz [c.f(z)] = c . (df/dz)
- d/dz zn = nzn-1
- d/dz (f.g) = f (dg/dz) + g (df/dz)
- d/dz (f/g) = [g (df/dz) – f (dg/dz)]/ g2
All these formulas are used to solve various problems in complex analysis.
Analytic Functions
A function f(z) is said to be analytic at a point z0 if f is differentiable not only at z, but also at every point in some neighbourhood of z0. Analytic functions are also called regular, holomorphic, or monogenic functions.
Also, check:Analytic functions
Harmonic Function
A function u(x, y) is said to be a harmonic function if it satisfies the Laplace equation. Also, the real and imaginary parts of an analytic function are harmonic functions.
Complex Integration
Suppose f(z) be a function of complex variable defined in a domain D and “c” be the closed curve in the domain D.
Let f(z) = u(x, y) + i v(x, y)
Here, z = x + iy
(or)
f(z) = u + iv and dz = dx + i dy
∫c f(z) dz = ∫c (u + iv) dz
= ∫c (u + iv) (dx + idy)
= ∫c (udx – vdy) +i ∫c (udy + vdx)
Here, ∫c f(z) dz is known as the contour integral.
Cauchy’s Integral Theorem
If f(z) is analytic function in a simply-connected region R, then ∫c f(z) dz = 0 for every closed contour c contained in R.
Converse:
If a function f(z) is continuous throughout the simple connected domain D and if ∫c f(z) dz = 0 for every closed contour c in D, then f(z) is analytic in D.
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