Zeigen Sie: F?r jede komplexe Zahl \(z=x+j y\) gilt: a) \(z+z^{*}=2 \cdot \operatorname{Re}(z)\) b) \(z-z^{*}=2 \mathrm{j} \cdot \operatorname{Im}(z)\)

Complex numbers are essential in engineering mathematics. They extend the number system beyond the real numbers and can be used to solve equations that have no real solutions. A complex number is generally represented as \(z = x + j y\), where \(x\) is the real part and \(y\) is the imaginary part. Here, \(j\) represents the imaginary unit, which satisfies \(j^2 = -1\).Complex numbers have various applications in fields like signal processing, control systems, and electromagnetics. They allow engineers to model and solve problems that involve oscillatory and wave-like behavior.
Examples of complex numbers:

• \(3 + 4j\)
• \(-2 - 5j\)
• \(7 + 0j\)

Working with complex numbers involves operations like addition, subtraction, multiplication, and division, very similar to real numbers but respecting the imaginary unit properties.

A complex conjugate of a complex number \(z\) is found by changing the sign of its imaginary part. If \(z = x + j y\), then its complex conjugate \(z^*\) is \(x - j y\). This operation is crucial in many applications such as simplifying complex fractions and in solving polynomial equations with complex coefficients.
For example:

• The complex conjugate of \(3 + 4j\) is \(3 - 4j\).
• The complex conjugate of \(-2 - 5j\) is \(-2 + 5j\).

Conjugates are particularly useful because:

• Adding a number and its conjugate gives twice the real part: \(z + z^* = 2x\).
• Subtracting the conjugate from the number gives twice the imaginary part times \(j\): \(z - z^* = 2jy\).

Every complex number \(z = x + j y\) consists of a real part \(x\) and an imaginary part \(y\). The real part can be represented as \(\operatorname{Re}(z)\) and the imaginary part as \(\operatorname{Im}(z)\). Identifying these parts separately is important to solve many problems in engineering mathematics.
For example:

• For \(z = 5 + 6j\), \(\operatorname{Re}(z) = 5\) and \(\operatorname{Im}(z) = 6\).
• For \(z = -3 - 4j\), \(\operatorname{Re}(z) = -3\) and \(\operatorname{Im}(z) = -4\).

It's useful to remember:

• The real part impacts the horizontal axis of the complex plane.
• The imaginary part impacts the vertical axis of the complex plane.

Complex numbers can be represented graphically on the complex plane, where the x-axis is the real part and the y-axis is the imaginary part.

In engineering, mathematics is a powerful tool, and complex numbers form a significant part of this toolkit. They are used in a variety of applications, including:

• Signal Processing: Analyzing and filtering signals using Fourier transforms.
• Control Systems: Designing and tuning controllers that manage the stability and response of mechanical systems.
• Electromagnetic Theory: Solving Maxwell's equations for electric and magnetic fields using phasors.

Complex numbers help in solving linear differential equations with constant coefficients and analyzing AC circuits using phasors.
Understanding the concepts of complex conjugates and the real and imaginary parts is fundamental for engineers to excel in these areas. Rendering complex signals into real and imaginary components allows treating multi-dimensional problems more manageably.
Using complex numbers efficiently in engineering requires a solid understanding of both theoretical concepts and practical applications.