Analytic Functions
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Table of Contents
Introduction
In the fascinating realm of complex functions, we explore a wide range of analytic functions that extend familiar real-number operations into the complex plane. This article delves into the definitions and properties of functions of a complex variable, covering topics from polynomial and rational functions to exponential, trigonometric, hyperbolic, and logarithmic functions. By examining these complex functions, you will gain insights into their structures, domains, and practical applications in both pure and applied mathematics.
Functions of a Complex Variable
Let S be a set of complex numbers. A function f defined on S is a rule that assigns toeach z in S a complex number w.
The number w is called the value of f at z and isdenoted by f (z), that is, w = f (z).
The set S is called the domain of definition of f.
Examples
Example-1
\(f (z) = z^{2}\)
In Cartesian Co-ordinates
Let \(z=x+iy \) then \[f(x+iy)=(x+iy)^2=x^2-y^2+i2xy \]
In Polar Co-ordinates
Let \(z=re^{i\theta} \) then \[f(re^{i\theta})=r^{2}e^{i2\theta} = r^{2} \cos 2\theta + ir^{2} \sin 2\theta \]
Domain of Definition
Example-1
Find the domain of definition of the function \[ f(z)=\frac{1}{z^2+1} \]
To find the domain of definition of the function \[ f(z)=\frac{1}{z^2+1}, \] we need to determine for which values of \( z \) the expression is defined. The function is undefined when its denominator is zero. So we must have \[ z^2+1\neq 0. \] Solving the equation \( z^2+1=0 \), which has the solutions \( z=i \) and \( z=-i \).
Therefore, the domain of definition of \( f(z)\) is \[ \{ z\in\mathbb{C} : z\neq i\text{ and }z\neq -i \}. \]
Example-2
Find the domain of definition of the function \[ f(z)=\operatorname{Arg}\left(\frac{1}{z}\right) \]
To find the domain of definition of \[ f(z)=\operatorname{Arg}\left(\frac{1}{z}\right) \]
The \(\operatorname{Arg}\) function is defined for any nonzero complex number. Since \(\frac{1}{z}\) will be nonzero as long as \(z\) is nonzero, the only restriction comes from the denominator in the fraction.
Thus, the domain of \( f(z) \) is \[ \{ z\in\mathbb{C} : z\neq 0 \}. \]
Example-3
Find the domain of definition of the function \[ f(z)=\frac{z}{z+\bar{z}} \]
To find the domain of definition of \[ f(z)=\frac{z}{z+\bar{z}}, \] we need to ensure that the expression is defined, which means the denominator must not be zero.
Notice that, \[ z+\bar{z}=2\operatorname{Re}(z). \] Thus, the denominator becomes zero when \[ \operatorname{Re}(z)=0. \]
Therefore the domain of definition \( f(z) \) is \[ \{ z\in\mathbb{C} : \operatorname{Re}(z)\neq 0 \}. \]
Example-4
Find the domain of definition of the function \[ f(z)=\frac{1}{1-|z|^{2}} \]
To determine the domain of definition for \[ f(z)=\frac{1}{1-|z|^2}, \] we need to ensure that the expression is defined, which means the denominator must not be zero.
So we require \[ 1-|z|^2 \neq 0\implies |z| \neq 1. \]
Therefore the domain of definition \( f(z) \) is\[ \{ z \in \mathbb{C} : |z| \neq 1 \}. \]
Example-4
Find the domain of definition of the function \[ f(z)=\frac{(2z+3)(z-1)}{z^{2}-2z+4} \]
To determine the domain of the function \[ f(z)=\frac{(2z+3)(z-1)}{z^2-2z+4}, \] we must identify the values of \( z \) for which the expression is defined. We require that \[ z^2-2z+4\neq 0 \implies z\neq\frac{2\pm2i\sqrt{3}}{2}\neq 1\pm i\sqrt{3}. \]
Therefore the domain of definition \( f(z) \) is \[ \{ z\in\mathbb{C} : z\neq 1\pm i\sqrt{3} \}. \]
1. Polynomial Functions
A polynomial function is defined by \[ w = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1}z + a_n = P(z), \] where \(a_0, a_1, \dots, a_n\) are complex constants with \(a_0 \neq 0\) and \(n\) is a positive integer (the degree of the polynomial). A transformation of the form \[ w = az + b \] is called a linear transformation.
2. Rational Algebraic Functions
A rational algebraic function is given by \[ w = \frac{P(z)}{Q(z)}, \] where \(P(z)\) and \(Q(z)\) are polynomials. In particular, the transformation \[ w = \frac{az+b}{cz+d} \] (with \(ad – bc \neq 0\)) is often called a bilinear or fractional linear transformation.
3. Exponential Functions
The complex exponential function is defined as \[ w = e^z = e^{x+iy} = e^x\Bigl(\cos y + i\sin y\Bigr), \] where \(e\) is the base of the natural logarithm. For a real positive number \(a\), we define \[ a^z = e^{z \ln a}. \] In particular, when \(a = e\), this definition reduces to the usual exponential function.
The exponential function satisfies properties similar to its real counterpart, for example: \[ e^{z_1} e^{z_2} = e^{z_1+z_2} \quad \text{and} \quad \bigl(e^{z_1}\bigr)^{z_2} = e^{z_1z_2}. \]
4. Trigonometric Functions
The complex trigonometric (or circular) functions are defined in terms of exponential functions as follows:
-
\(\displaystyle \sin z = \frac{e^{iz} – e^{-iz}}{2i}\)
Domain of definition: \(\{ z\in\mathbb{C}\}\). -
\(\displaystyle \cos z = \frac{e^{iz} + e^{-iz}}{2}\)
Domain of definition: \(\{ z\in\mathbb{C}\}\). -
\(\displaystyle \sec z = \frac{1}{\cos z} = \frac{2}{e^{iz} + e^{-iz}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq \frac{\pi}{2}+n\pi,\ n\in\mathbb{Z}\}\). -
\(\displaystyle \csc z = \frac{1}{\sin z} = \frac{2i}{e^{iz} – e^{-iz}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq n\pi,\ n\in\mathbb{Z}\}\). -
\(\displaystyle \tan z = \frac{\sin z}{\cos z} = \frac{e^{iz} – e^{-iz}}{i\bigl(e^{iz} + e^{-iz}\bigr)}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq \frac{\pi}{2}+n\pi,\ n\in\mathbb{Z}\}\). -
\(\displaystyle \cot z = \frac{\cos z}{\sin z} = \frac{i\bigl(e^{iz} + e^{-iz}\bigr)}{e^{iz} – e^{-iz}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq n\pi,\ n\in\mathbb{Z}\}\).
Some fundamental identities include: \[ \sin^2 z + \cos^2 z = 1,\quad 1+\tan^2 z = \sec^2 z,\quad 1+\cot^2 z = \operatorname{cosec}^2 z. \]
The addition formulas are:
- \(\displaystyle \sin(z_1+z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2\)
- \(\displaystyle \cos(z_1+z_2) = \cos z_1 \cos z_2 – \sin z_1 \sin z_2\)
- \(\displaystyle \tan(z_1+z_2) = \frac{\tan z_1 + \tan z_2}{1 – \tan z_1 \tan z_2}\)
5. Hyperbolic Functions
The hyperbolic functions are defined by:
-
\(\displaystyle \sinh z = \frac{e^z – e^{-z}}{2}\)
Domain of definition: \(\{ z\in\mathbb{C}\}\). -
\(\displaystyle \cosh z = \frac{e^z + e^{-z}}{2}\)
Domain of definition: \(\{ z\in\mathbb{C}\}\). -
\(\displaystyle \operatorname{sech} z = \frac{1}{\cosh z} = \frac{2}{e^z + e^{-z}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq i\frac{\pi}{2}(2n+1),\ n\in\mathbb{Z}\}\). -
\(\displaystyle \operatorname{cosech} z = \frac{1}{\sinh z} = \frac{2}{e^z – e^{-z}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq i\pi n,\ n\in\mathbb{Z}\}\). -
\(\displaystyle \tanh z = \frac{\sinh z}{\cosh z} = \frac{e^z – e^{-z}}{e^z + e^{-z}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq i\frac{\pi}{2}(2n+1),\ n\in\mathbb{Z}\}\). -
\(\displaystyle \coth z = \frac{\cosh z}{\sinh z} = \frac{e^z + e^{-z}}{e^z – e^{-z}}\)
Domain of definition: \(\{ z\in\mathbb{C} : z \neq i \pi n,\ n\in\mathbb{Z}\}\).
They satisfy the identities: \[ \cosh^2 z – \sinh^2 z = 1,\quad 1 – \tanh^2 z = \operatorname{sech}^2 z,\quad \coth^2 z – 1 = \operatorname{cosech}^2 z. \]
The addition formulas are:
- \(\displaystyle \sinh(z_1+z_2) = \sinh z_1 \cosh z_2 + \cosh z_1 \sinh z_2\)
- \(\displaystyle \cosh(z_1+z_2) = \cosh z_1 \cosh z_2 + \sinh z_1 \sinh z_2\)
- \(\displaystyle \tanh(z_1+z_2) = \frac{\tanh z_1 + \tanh z_2}{1+\tanh z_1 \tanh z_2}\)
The following relations connect the trigonometric and hyperbolic functions:
- \(\displaystyle \sin(iz) = i\sinh z,\quad \cos(iz) = \cosh z,\quad \tan(iz) = i\tanh z,\)
- \(\displaystyle \sinh(iz) = i\sin z,\quad \cosh(iz) = \cos z,\quad \tanh(iz) = i\tan z.\)
6. Logarithmic Functions
If \[ z = e^w, \] then we write \[ w = \ln z. \] In general, since \[ z = re^{iu}, \] the logarithm is given by \[ \ln z = \ln r + i\Bigl(u + 2k\pi\Bigr),\quad k\in\mathbb{Z}. \] The principal value is obtained by choosing \(0 \le u \lt 2\pi\).
For a real base \(a\) with \(a>0\) and \(a\neq 1\), if \[ z = a^w, \] then \[ w = \log_a z, \] and since \[ z = e^{w\ln a}, \] it follows that \[ \log_a z = \frac{\ln z}{\ln a}. \]
7. Inverse Trigonometric Functions
The inverse trigonometric (or circular) functions can be expressed in terms of logarithms. For example,
-
\(\displaystyle \sin^{-1}z = \frac{1}{i}\ln\Bigl(iz+\sqrt{1-z^2}\,\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\notin (-\infty,-1]\cup[1,\infty)\}\). -
\(\displaystyle \cos^{-1}z = \frac{1}{i}\ln\Bigl(z+\sqrt{z^2-1}\,\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\notin (-\infty,-1]\cup[1,\infty)\}\). -
\(\displaystyle \tan^{-1}z = \frac{1}{2i}\ln\Bigl(\frac{1+iz}{1-iz}\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\neq i \text{ and } z\neq -i\}\). -
\(\displaystyle \cot^{-1}z = \frac{1}{2i}\ln\Bigl(\frac{z+i}{z-i}\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\neq i \text{ and } z\neq -i\}\). -
\(\displaystyle \operatorname{cosec}^{-1}z = \frac{1}{i}\ln\Bigl(\frac{i+\sqrt{z^2-1}}{z}\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\neq 0\}\). -
\(\displaystyle \operatorname{sec}^{-1}z = \frac{1}{i}\ln\Bigl(\frac{1}{z}+\sqrt{\frac{1}{z^2}-1}\Bigr),\)
Domain of definition: \(\{\,z\in\mathbb{C} : z\neq 0\}\).
8. Inverse Hyperbolic Functions
The inverse hyperbolic functions are given by:
-
\(\displaystyle \sinh^{-1}z = \ln\Bigl(z+\sqrt{z^2+1}\,\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} \}\) (with the standard branch cut chosen between the branch points \(z=-i\) and \(z=i\)). -
\(\displaystyle \cosh^{-1}z = \ln\Bigl(z+\sqrt{z^2-1}\,\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} : z\notin (-\infty,1] \}\). -
\(\displaystyle \tanh^{-1}z = \frac{1}{2}\ln\Bigl(\frac{1+z}{1-z}\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} : z\notin (-\infty,-1]\cup[1,\infty) \}\). -
\(\displaystyle \coth^{-1}z = \frac{1}{2}\ln\Bigl(\frac{z+1}{z-1}\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} : z\neq \pm 1 \}\). -
\(\displaystyle \operatorname{sech}^{-1}z = \ln\Bigl(\frac{1}{z}+\sqrt{\frac{1}{z^2}-1}\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} : z\neq 0 \}\). -
\(\displaystyle \operatorname{cosech}^{-1}z = \ln\Bigl(\frac{1}{z}+\sqrt{\frac{1}{z^2}+1}\Bigr),\)
Domain of definition: \(\{ z\in\mathbb{C} : z\neq 0 \}\).
9. Exponentiation with Complex Exponents
For a complex exponent \(a\), the function \(z^a\) is defined by \[ z^a = e^{a\ln z}. \] More generally, if \(f(z)\) and \(g(z)\) are functions, then \[ f(z)^{g(z)} = e^{\,g(z)\ln f(z)}, \] and these functions are generally multiple‐valued.
10. Algebraic and Transcendental Functions
If \(w\) is a solution of the polynomial equation \[ P_0(z)w^n + P_1(z)w^{n-1} + \cdots + P_{n-1}(z)w + P_n(z) = 0, \] where \(P_0(z) \neq 0\) and \(P_1(z), \dots, P_n(z)\) are polynomials in \(z\), then the function \[ w = f(z) \] is called an algebraic function of \(z\).
A function which cannot be expressed as a polynomial equation is said to be a transcendental function. The logarithmic, trigonometric, and hyperbolic functions and their inverses are examples of transcendental functions.
Conclusions
In summary, the study of complex functions reveals a rich tapestry of mathematical structures that extend classical real analysis into the broader and more intricate complex plane. Whether you are examining polynomial and rational functions or exploring the elegant formulations of exponential, trigonometric, hyperbolic, and logarithmic functions, each topic contributes to a comprehensive understanding of analytic functions.
Through the detailed definitions and properties presented, this article has showcased the versatility and depth of functions of a complex variable. The interplay between various functions—demonstrated by their multiple representations and domains—illustrates the inherent beauty of complex analysis. As you continue to study these concepts, you will find that the principles discussed not only serve as foundational elements in mathematics but also have profound implications in physics, engineering, and other scientific disciplines.
Embrace these ideas and let the intricate world of complex functions inspire further exploration and discovery.
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FAQs
Group Theory
- What is a group in group theory?
A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for every element.
- What are the main properties of a group?
The four main properties are:
- Closure: The result of the operation on any two elements of the group is also in the group.
- Associativity: The group operation is associative.
- Identity: There exists an element that does not change other elements when used in the operation.
- Invertibility: Every element has an inverse that, when combined with the element, yields the identity.
- What is the identity element in a group?
The identity element is a unique element in the group that, when combined with any other element using the group operation, leaves that element unchanged. It is commonly denoted by e or 1.
- What is an abelian group?
An abelian group is one in which the binary operation is commutative. This means for any two elements a and b in the group, a · b = b · a
- What is a subgroup?
A subgroup is a subset of a group that is itself a group under the same binary operation. It must satisfy the group properties: closure, associativity, identity, and inverses.
- What is a normal subgroup and how does it relate to quotient groups?
A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group. This means for every element n in the normal subgroup N and every element g in the group, gng⁻¹ is still in N. Normal subgroups allow the construction of quotient groups, where the group is partitioned into cosets of the normal subgroup.
- What are group homomorphisms?
A group homomorphism is a function between two groups that preserves the group operation. This means if f: G → H is a homomorphism and a, b are elements of G, then f(a · b) = f(a) · f(b) in H.
- What is Lagrange’s theorem in group theory?
Lagrange's theorem states that for any finite group, the order (number of elements) of every subgroup divides the order of the entire group. This theorem is a fundamental result in the study of finite groups.
- What is Cayley’s theorem?
Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group. This implies that every group can be represented as a group of permutations acting on a set.
- How is group theory applied in other fields?
Group theory has applications in many fields including:
- Physics: Describing symmetries and conservation laws.
- Chemistry: Analyzing molecular symmetry and chemical bonding.
- Cryptography: Underlying structures in cryptographic systems.
- Mathematics: Foundational in algebra, geometry, and number theory.
