Problem 2 - Arbeitsbuch Mathematik für Ingenieure

Question

Berechnen Sie die Kurvenintegrale $$ \oint_{K} \frac{1}{(z+i)(z-2)} d z $$ über die Kreise \(K=K_{1.5}(0), K=K_{2}(3)\) und \(K=K_{0.5}(0)\).

Step-by-Step Solution

Answer

For \( K_{1.5}(0) \), the integral is \( -\frac{4\pi}{5} - \frac{2\pi i}{5} ); for \( K_{2}(3) \), it is \( \frac{4\pi i}{5} + \frac{2\pi}{5} \); for \( K_{0.5}(0) \), it is zero.

Step 1: Define the Contour Integrals

First, identify the provided integrals and their corresponding contours. The function given is \[ f(z) = \frac{1}{(z + i)(z - 2)} \] and we need to evaluate this over the circles \( K_{1.5}(0) \), \( K_{2}(3) \), and \( K_{0.5}(0) \).

Step 2: Identify the Singularities

Determine where the function has singularities by finding the values of \(z\) that make the denominator zero. These are at \( z = -i \) and \( z = 2 \).

Step 3: Analyze Contour for \( K_{1.5}(0) \)

The circle \( K_{1.5}(0) \) is centered at \( 0 \) with radius \( 1.5 \). Check if any singularities lie within this contour: the point \( z = -i \) (which is at distance \( 1 \) from the origin) lies within the circle, while \( z = 2 \) does not. Since only one singularity \( z = -i \) lies inside, we use the residue theorem: \[ \oint_{K_{1.5}(0)} f(z) \, dz = 2\pi i \cdot \text{Res}(f,-i) \].

Step 4: Calculate the Residue for \( z = -i \)

To calculate the residue of \( f(z) \) at \( z = -i \), use \[ \text{Res}(f, -i) = \lim_{z \to -i} (z + i) f(z) = \lim_{z \to -i} \frac{1}{z - 2} = \frac{1}{-i - 2} = \frac{1}{-2 - i}. \]

Step 5: Simplify the Residue

Simplify \( \frac{1}{-2 - i} \) by multiplying numerator and denominator by the conjugate: \[ \frac{1}{-2 - i} = \frac{1}{(-2 - i)} \cdot \frac{-2 + i}{-2 + i} = \frac{-2 + i}{4 + 1} = \frac{-2 + i}{5} = \frac{-2}{5} + \frac{i}{5}. \]

Step 6: Finalize the Contour Integral for \( K_{1.5}(0) \)

Using the residue, the contour integral is: \[ \oint_{K_{1.5}(0)} f(z) \, dz = 2\pi i \cdot \left( \frac{-2}{5} + \frac{i}{5} \right) = \frac{2\pi i \cdot -2}{5} + \frac{2\pi i \cdot i}{5} = -\frac{4\pi i}{5} + \frac{2\pi (-1)}{5} = -\frac{4\pi}{5} - \frac{2\pi i}{5}. \]

Step 7: Analyze Contour for \( K_{2}(3) \)

The circle \( K_{2}(3) \) is centered at \( 3 \) with radius \( 2 \). Calculate the distances: \( |3 - (-i)| = \sqrt{10} \approx 3.16 \) and \( |3 - 2| = 1 \). 'z = 2' is inside the contour, while \( z = -i \) is outside. Thus, only \( z = 2 \) lies inside. So use: \[ \oint_{K_{2}(3)} f(z) \, dz = 2\pi i \cdot \text{Res}(f, 2). \]

Step 8: Calculate Residue for \( z = 2 \)

Using the same technique as before: \[ \text{Res}(f,2) = \lim_{z \to 2} (z - 2) f(z) = \lim_{z \to 2} \frac{1}{z + i} = \frac{1}{2 + i}. \] Simplify as: \[ \frac{1}{2 + i} \cdot \frac{2 - i}{2 - i} = \frac{2 - i}{4 + 1} = \frac{2 - i}{5} = \frac{2}{5} - \frac{i}{5}. \]

Step 9: Finalize the Contour Integral for \( K_{2}(3) \)

Using the residue, the contour integral is :\[ \oint_{K_{2}(3)} f(z) \, dz = 2\pi i \cdot \left( \frac{2}{5} - \frac{i}{5} \right) = \frac{4\pi i}{5} - \frac{2\pi (-1)}{5} = \frac{4\pi i}{5} + \frac{2\pi}{5} = \frac{4\pi i}{5} + \frac{2\pi}{5}. \]

Step 10: Analyze Contour for \( K_{0.5}(0) \)

The circle \( K_{0.5}(0) \) is centered at \( 0 \) with radius \( 0.5 \). Neither of the singularities \( z = -i \) and \( z = 2 \) lie within this contour. Therefore, no singularities are enclosed, implying: \[ \oint_{K_{0.5}(0)} f(z) \, dz = 0. \]

Key Concepts

Residuenrechnung

Residuenrechnung, also known as the residue theorem, is a powerful tool in complex analysis. It allows us to evaluate contour integrals of meromorphic functions by summing the residues of the function's singularities within the contour. The residue of a function at a singularity is essentially the coefficient of the \(\frac{1}{z - z_0}\) term in its Laurent series expansion.

Here's a concise breakdown:

• Identify singularities (poles) within the contour.
• Calculate the residue at each pole.
• Apply the residue theorem: \[ \oint_{C} f(z) \, dz = 2\pi i \, \sum \text{Res}(f, z_i). \]

For example, in our integral, \[ f(z) = \frac{1}{(z+i)(z-2)}, \] the singularities are at \( z = -i \) and \( z = 2 \). Once we identify which contour encompasses these singularities, we calculate their residues and sum them up.

Komplexe Funktionen

Komplexe Funktionen, or complex functions, are functions that take complex numbers as inputs and provide complex numbers as outputs. These functions can be incredibly powerful in both pure and applied mathematics.

Some key properties include:

• Holomorphy: A function is holomorphic if it is complex differentiable at every point in its domain.
• Analyticity: If a function is holomorphic, it can be locally represented by a convergent power series (its Taylor series).

In the given problem, the function is \( f(z) = \frac{1}{(z+i)(z-2)} \). This function is meromorphic, having isolated singularities (poles) at \( z = -i \) and \( z = 2 \). By studying these complex functions, we can leverage theorems from complex analysis, such as the residue theorem, to simplify and solve integrals.

Konturanalyse

Konturanalyse, or contour analysis, involves studying the properties of a complex function along a closed curve, also known as a contour. This technique is crucial in complex analysis.

To analyze a contour, consider these steps:

• Identify the contour: Understand the shape and position of the curve in the complex plane.
• Singularity check: Determine which singularities of the function lie inside the contour.
• Application of theorems: Use theorems like the residue theorem to evaluate the integral around the contour.

In the given problem, the contours are circles of different radii:
1. \( K_{1.5}(0) \) is a circle centered at \( 0 \) with radius \( 1.5 \), which contains \( z = -i \) within its interior.
2. \( K_{2}(3) \) is a circle centered at \( 3 \) with radius \( 2 \), containing \( z = 2 \).
3. \( K_{0.5}(0) \) is a circle centered at \( 0 \) with radius \( 0.5 \), which does not enclose any singularities.
By carefully analyzing the contours, we can calculate the contour integrals using appropriate theorems and techniques.