4.1 Exponential and logarithmic functions

Complex exponentials and logarithms are key players in complex analysis. They extend real-valued functions to the complex plane, bringing new properties and challenges. These functions are inverses of each other but behave differently in the complex world.

Understanding complex exponentials and logarithms is crucial for tackling advanced topics in complex analysis. They help define complex powers, trigonometric functions, and hyperbolic functions, opening doors to solving intricate problems in various fields of mathematics and physics.

Complex Exponential Functions

Definition and Evaluation

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• The complex exponential function is defined as $e^{z} = e^{x+iy} = e^x(\cos(y) + i\sin(y))$, where $z = x + iy$ is a complex number, $x$ is the real part, and $y$ is the imaginary part
• Express the complex exponential function using Euler's formula: $e^{iy} = \cos(y) + i\sin(y)$, where $i$ is the imaginary unit and $y$ is a real number
• The complex exponential function is periodic with period $2\pi i$, meaning $e^{z} = e^{z+2\pi i}$ for any complex number $z$

Properties of the Complex Exponential Function

• The absolute value (modulus) of the complex exponential function is given by $|e^{z}| = e^x$, where $x$ is the real part of the complex number $z$
  • For example, if $z = 2 + 3i$, then $|e^{z}| = e^2 \approx 7.39$
• The argument (angle) of the complex exponential function is given by $\arg(e^{z}) = y + 2\pi k$, where $y$ is the imaginary part of the complex number $z$ and $k$ is an integer
  • For instance, if $z = 1 + \pi i$, then $\arg(e^{z}) = \pi + 2\pi k$, where $k \in \mathbb{Z}$
• The complex exponential function maps vertical lines in the complex plane to circles centered at the origin
• The derivative of the complex exponential function is itself, i.e., $\frac{d}{dz}e^{z} = e^{z}$

Properties of Complex Logarithms

Definition and Multi-Valued Nature

• The complex logarithm, denoted as $\log(z)$ or $\ln(z)$, is the inverse function of the complex exponential function $e^{z}$
• The complex logarithm is a multi-valued function, meaning that for a given complex number $z$, there are infinitely many values of $\log(z)$ that differ by integer multiples of $2\pi i$
  • For example, if $z = 1$, then $\log(1) = 2\pi i k$, where $k \in \mathbb{Z}$
• The principal value of the complex logarithm, denoted as $\operatorname{Ln}(z)$ or $\operatorname{Log}(z)$, is the value with the imaginary part lying in the interval $(-\pi, \pi]$

Properties and Expressions

• Express the complex logarithm as $\log(z) = \ln|z| + i(\arg(z) + 2\pi k)$, where $|z|$ is the modulus of $z$, $\arg(z)$ is the principal argument of $z$, and $k$ is an integer
• The complex logarithm is not defined for $z = 0$ and has a branch cut along the negative real axis
• The complex logarithm satisfies the properties of logarithms, such as:
  • $\log(zw) = \log(z) + \log(w)$ for complex numbers $z$ and $w$
  • $\log(z^n) = n\log(z)$ for complex number $z$ and integer $n$
• The complex logarithm maps circles centered at the origin to vertical lines (with the exception of the negative real axis)
• The derivative of the complex logarithm is the reciprocal of the variable, i.e., $\frac{d}{dz}\log(z) = \frac{1}{z}$

Solving Complex Equations

Solving Equations with Complex Exponentials

• To solve equations involving complex exponentials, use the properties of exponents and Euler's formula to simplify the equation and isolate the variable
  • For instance, solve $e^{2z} = 1$ by taking the natural logarithm of both sides: $2z = \ln(1) + 2\pi i k$, where $k \in \mathbb{Z}$, then $z = \pi i k$
• Be aware of the periodicity of the complex exponential function when solving equations, as there may be infinitely many solutions

Solving Equations with Complex Logarithms

• When solving equations with complex logarithms, ensure that the arguments of the logarithms are non-zero complex numbers and that the equation is consistent with the branch cuts and principal values of the logarithms
• Use the properties of logarithms, such as $\log(zw) = \log(z) + \log(w)$ and $\log(z^n) = n\log(z)$, to simplify equations involving complex logarithms
  • For example, solve $\log(z^2) = 2 + 4\pi i$ by using the property $\log(z^2) = 2\log(z)$: $2\log(z) = 2 + 4\pi i$, then $\log(z) = 1 + 2\pi i$, and finally $z = e^{1+2\pi i}$
• Consider the multi-valued nature of the complex logarithm when solving equations, as there may be multiple solutions or no solutions depending on the branch cut and principal value chosen

Solving Equations with Both Complex Exponentials and Logarithms

• When an equation involves both complex exponentials and logarithms, apply the inverse relationship between the functions, i.e., $e^{\log(z)} = z$ and $\log(e^{z}) = z$, to simplify the equation
  • For instance, solve $e^{\log(z)} = 2$ by using the inverse relationship: $z = 2$
• Combine the techniques for solving equations with complex exponentials and logarithms when faced with equations involving both functions

Complex Exponentials vs Logarithms

Inverse Relationship

• The complex exponential function $e^{z}$ and the complex logarithm $\log(z)$ are inverse functions, meaning that:
  • $e^{\log(z)} = z$ for non-zero complex numbers $z$
  • $\log(e^{z}) = z$ for all complex numbers $z$
• This inverse relationship allows for simplifying expressions and solving equations involving both complex exponentials and logarithms

Contrasting Properties

• The complex exponential function is a periodic function with period $2\pi i$, while the complex logarithm is a multi-valued function with a branch cut along the negative real axis
• The complex exponential function maps vertical lines in the complex plane to circles centered at the origin, while the complex logarithm maps circles centered at the origin to vertical lines (with the exception of the negative real axis)

Importance in Complex Analysis

• The complex exponential and logarithmic functions play a crucial role in complex analysis, as they are used to define and study:
  • Complex powers: $z^w = e^{w \log(z)}$ for complex numbers $z$ and $w$
  • Complex trigonometric functions: $\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$ and $\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$
  • Complex hyperbolic functions: $\cosh(z) = \frac{e^{z} + e^{-z}}{2}$ and $\sinh(z) = \frac{e^{z} - e^{-z}}{2}$
• Understanding the properties and behavior of complex exponentials and logarithms is essential for advanced topics in complex analysis, such as contour integration and residue theory