Problem 8 - Complex Analysis

Question

For \(z \in \mathbb{C} \backslash\\{(k+1 / 2) \pi ; k \in \mathbb{Z}\\}\) let $$ \tan z:=\frac{\sin z}{\cos z} $$ and for \(z \in \mathbb{C} \backslash\\{k \pi ; k \in \mathbb{Z}\\}\) let $$ \cot z:=\frac{\cos z}{\sin z} . $$ Show: $$ \begin{array}{ll} \tan z=\frac{1}{\mathrm{i}} \frac{\exp (2 \mathrm{i} z)-1}{\exp (2 \mathrm{i} z)+1}, & \tan (z+\pi / 2)=-\cot z \\ \tan (-z)=-\tan z \\ \cot z=\mathrm{i} \frac{\exp (2 \mathrm{i} z)+1}{\exp (2 \mathrm{i} z)-1}, & \tan z=\tan (z+\pi) \end{array} $$ $$ \begin{aligned} &\tan z=\cot z-2 \cot (2 z) \\ &\cot (z+\pi)=\cot z. \end{aligned} $$

Step-by-Step Solution

Answer

The identities hold as shown using exponential forms of sine and cosine and the periodic properties of complex tangent and cotangent functions.

Step 1: Understanding the Definitions

We are given the definitions of the tangent and cotangent functions in terms of sine and cosine for complex numbers. For \( \tan z = \frac{\sin z}{\cos z} \) and \( \cot z = \frac{\cos z}{\sin z} \). We need to use these definitions along with exponential forms of sine and cosine.

Step 2: Rewriting Using Exponential Functions

Using Euler's formula, \( \sin z = \frac{ \exp(\mathrm{i} z) - \exp(-\mathrm{i} z) }{2\mathrm{i}} \) and \( \cos z = \frac{ \exp(\mathrm{i} z) + \exp(-\mathrm{i} z) }{2} \). Substitute these into \( \tan z = \frac{\sin z}{\cos z} \).

Step 3: Simplify Tan Function with Exponentials

After substituting, \( \tan z = \frac{ \frac{ \exp(\mathrm{i} z) - \exp(-\mathrm{i} z) }{2\mathrm{i}} }{ \frac{ \exp(\mathrm{i} z) + \exp(-\mathrm{i} z) }{2} } = \frac{1}{\mathrm{i}} \cdot \frac{\exp(2\mathrm{i}z) - 1}{\exp(2\mathrm{i}z) + 1} \). This confirms the given expression for \( \tan z \).

Step 4: Tangent Identity Proofs

For \( \tan (z+\pi/2) \), compute \( \tan z = \frac{1}{\mathrm{i}} \cdot \frac{\exp[2\mathrm{i}(z+\pi/2)] - 1}{\exp[2\mathrm{i}(z+\pi/2)] + 1} = -\cot z \) from the properties of exponential functions.

Step 5: Symmetry in Tangent Function

\( \tan(-z) = \frac{1}{\mathrm{i}} \cdot \frac{\exp(-2\mathrm{i}z) - 1}{\exp(-2\mathrm{i}z) + 1} = -\tan(z) \), which uses the fact that \( \exp(-x) = 1/\exp(x) \). This verifies \( \tan(-z) = -\tan z \).

Step 6: Express Cotangent with Exponentials

For \( \cot z = \frac{\cos z}{\sin z} = \mathrm{i} \cdot \frac{\exp(2\mathrm{i}z) + 1}{\exp(2\mathrm{i}z) - 1} \), confirming the given formula.

Step 7: Periodicity of Tangent

The property \( \tan z = \tan(z+\pi) \) follows from tangent's periodic nature; applying transformations such as \( \tan(z+\pi) = -\tan z \) and \( -\tan(-z) = \tan z \).

Step 8: Cotangent Function with Shift

For \( \cot(z+\pi) = \cot z \), use the periodicity and symmetry \( \cot(z+\pi) = \frac{\cos(z+\pi)}{\sin(z+\pi)} = \frac{-\cos(z)}{-\sin(z)} = \cot z \).

Key Concepts

Complex Functions

Complex functions are central to complex analysis, a field that explores mathematical concepts extended to complex numbers. Complex numbers are of the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). Complex functions like \( \tan z \) and \( \cot z \) extend the familiar trigonometric functions beyond the real number system.
In this context, it is important to remember that complex functions can exhibit behaviors unlike their real counterparts. For example, they can be defined in terms of other complex functions and demonstrate different periodic or symmetrical properties that require deeper exploration.
Understanding these properties is vital, especially when dealing with singularities or certain points where the function does not behave typically, such as within the given exercise where exclusions are mentioned for \( \tan z \) and \( \cot z \). These exceptions alert us to potential undefined behavior in complex functions.

Trigonometric Functions

Trigonometric functions, when extended to the complex plane, reveal fascinating properties and applications that transcend their original geometric interpretations. Common trigonometric functions like sine, cosine, tangent, and cotangent relate to exponential functions through Euler's Formula, \( e^{ix} = \cos x + i \sin x \).
For example, the formulas used in the exercise for \( \tan z = \frac{\sin z}{\cos z} \) and \( \cot z = \frac{\cos z}{\sin z} \) take on new meanings as complex functions. They are often expressed using exponential terms for simplification and analysis. This reveals hidden symmetries and relationships, such as the identities:

• \( \tan (z + \pi/2) = -\cot z \)
• \( \cot (z+\pi) = \cot z \)

These highlight their periodic and symmetrical nature, essential for understanding the full scope of complex trigonometric behavior.

Exponential Functions

Exponential functions extend elegantly into the complex domain, offering a bridge between trigonometric functions and complex analysis. Euler's Formula \( e^{ix} = \cos x + i \sin x \) is particularly useful in this transition, providing a means to express trigonometric functions using exponentials.
For instance, the tangent and cotangent functions can be expressed in terms of exponential forms, as seen in the exercise. When transformed through exponentials, they appear as:

• \( \tan z = \frac{1}{i} \cdot \frac{e^{2iz} - 1}{e^{2iz} + 1} \)
• \( \cot z = i \cdot \frac{e^{2iz} + 1}{e^{2iz} - 1} \)

These expressions help to simplify calculations and derive properties such as symmetry and periodicity more efficiently. The interplay between exponential and trigonometric functions opens up advanced pathways in complex function analysis.

Periodic Functions

Periodic functions exhibit a repeating pattern at fixed intervals, a property very familiar with basic trigonometric functions like sine and cosine. In the complex plane, these periodic characteristics extend with additional nuances.
Functions such as \( \tan z \) and \( \cot z \) demonstrate specific periodic properties:

• \( \tan z = \tan(z + \pi) \)
• \( \cot z = \cot(z + \pi) \)

These expressions indicate that they repeat their values after a period of \( \pi \), a key characteristic borrowed from their real counterparts.
Periodic functions in complex analysis can provide insight into oscillatory behaviors and are fundamental in studying waves, signals, and other repeating phenomena in various applied fields. Recognizing and understanding these periodic properties allows for deeper analysis and application of complex functions.