Question

Zeigen Sie die Beziehungen \(\cosh ^{2} z-\sinh ^{2} z=1 \quad \sinh 2 z=2 \sinh z \cosh z\) \(\cosh (z+w)=\cosh z \cosh w+\sinh z \sinh w\) für die komplexen Hyperbelfunktionen.

Short answer

The identities are true for complex hyperbolic functions through expansions and simplifications based on their definitions.

Step-by-step solution

Understand the First Identity

First, we need to prove that \(\cosh^2 z - \sinh^2 z = 1\). This identity is analogous to the Pythagorean identity for trigonometric functions. The hyperbolic cosine and sine are defined as \(\cosh z = \frac{e^z + e^{-z}}{2}\) and \(\sinh z = \frac{e^z - e^{-z}}{2}\).

Expand the Terms

Substitute the definitions into the identity: \(\cosh^2 z = \left(\frac{e^z + e^{-z}}{2}\right)^2\) and \(\sinh^2 z = \left(\frac{e^z - e^{-z}}{2}\right)^2\).

Simplify Each Side

Compute \(\cosh^2 z\) as \(\frac{e^{2z} + 2 + e^{-2z}}{4}\) and \(\sinh^2 z\) as \(\frac{e^{2z} - 2 + e^{-2z}}{4}\). Subtract \(\sinh^2 z\) from \(\cosh^2 z\) to obtain \(\frac{4}{4} = 1\).

Proving the Second Identity

For \(\sinh 2z = 2 \sinh z \cosh z\), use \(\sinh 2z = \frac{e^{2z} - e^{-2z}}{2}\).Furthermore, \(2 \sinh z \cosh z = 2\left( \frac{e^z - e^{-z}}{2} \right)\left( \frac{e^z + e^{-z}}{2} \right)\) \(= \frac{e^{2z} - e^{-2z}}{2}\).

Proving the Third Identity

Demonstrate \(\cosh(z+w) = \cosh z \cosh w + \sinh z \sinh w\) using the definitions. Expand the left: \(\cosh(z+w) = \frac{e^{z+w} + e^{-(z+w)}}{2}\).This expands to \(\frac{(e^z e^w + e^{-z} e^{-w}) + (e^{-z} e^w + e^z e^{-w})}{2}\), which matches \(\cosh z \cosh w + \sinh z \sinh w\).

Conclusion

Summarize that each identity holds true by expanding and simplifying using definitions of the hyperbolic functions and algebraic operations.

Key concepts

Complex Analysis

Complex analysis is a branch of mathematics that deals with complex numbers and their functions.
It provides powerful techniques for solving problems related to calculus or any analysis involving continuous functions.
In complex analysis, functions often have real and imaginary components.With hyperbolic functions, complex analysis becomes an indispensable tool.
This is because hyperbolic functions, like \(\cosh(z)\) and \(\sinh(z)\), are defined using exponential functions involving complex variables.
Here's why complex analysis is useful:

• It allows us to study the properties and behavior of hyperbolic functions on the complex plane.
• Complex analysis helps prove identities and theorems by providing a structured approach to manipulate complex expressions.
• Using Euler's formula, one can translate trigonometric functions into exponential form, easing many calculations.

By understanding complex analysis, you gain the tools to tackle a broader range of mathematical problems efficiently.
It simplifies computations and aids in finding solutions to complex equations that might be challenging otherwise.

Mathematical Identities

Mathematical identities are equations that are true for all values of the variables involved.
Identifying these relationships is crucial for simplifying equations and solving complex problems.
In the context of hyperbolic functions, these identities often mirror trigonometric properties seen in triangles and circles.One essential identity for hyperbolic functions is the hyperbolic Pythagorean identity: \[ \cosh^2(z) - \sinh^2(z) = 1 \]. This identity is akin to the familiar trigonometric Pythagorean identity: \( \cos^2(x) + \sin^2(x) = 1 \).Why are these identities important?

• They allow us to simplify expressions that include hyperbolic functions.
• Identities can help solve complex equations by reducing or transforming terms.
• They provide insights into the nature of hyperbolic functions, revealing their similarities to trigonometric counterparts.

Understanding and applying these identities can lead to more efficient problem-solving techniques in many areas of mathematics.
They are fundamental tools used in calculus, differential equations, and further studies in theoretical physics.

Hyperbolic Trigonometric Functions

Hyperbolic trigonometric functions are analogs of regular trigonometric functions, but instead of dealing with circular geometry, they relate to hyperbolas.
These functions include hyperbolic sine \( \sinh(z) \) and hyperbolic cosine \( \cosh(z) \).Hyperbolic sine and cosine are defined using exponential functions:

• \( \cosh(z) = \frac{e^z + e^{-z}}{2} \)
• \( \sinh(z) = \frac{e^z - e^{-z}}{2} \)

These definitions show how hyperbolic functions can be seen as compositions of exponential functions with real or complex arguments.What makes hyperbolic functions special?

• They have properties similar to trigonometric functions, forming identities like \( \cosh^2(z) - \sinh^2(z) = 1 \).
• They appear naturally in certain types of differential equations and physical systems, especially in relation to hyperbolic geometry.
• Hyperbolic functions can describe catenaries (curves formed by a hanging chain) and can model certain hyperbolic movements or growth patterns.

Grasping the concept of hyperbolic trigonometric functions broadens your understanding of mathematical phenomena.
They provide a different perspective and toolkit for exploring functions that describe both real-world and theoretical models.