Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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How do I parameterize the portion from Rei pi/4 to 0?  

The image above displays a graph depicting the path of a contour in the complex plane. The contour starts at the origin (0, 0), moves linearly to the point represented by \(Re^{i\frac{\pi}{4}}\), and then follows a curved path back to the point \(R\) on the real axis. ### Graph Explanation: 1. **Axes**: - The horizontal axis is labeled \(R\) and represents the real part of the complex plane. - The vertical axis is not labeled but typically represents the imaginary part of the complex plane. 2. **Path Description**: - The contour begins at the origin, indicated by the point \(0\) at the intersection of the two axes. - It then proceeds linearly to the point labeled \(Re^{i\frac{\pi}{4}}\). This point is at a distance \(R\) from the origin and makes an angle of \( \frac{\pi}{4}\) (or 45 degrees) with the positive real axis. - The path then follows a curved line back to the point \(R\) on the real axis. This curved part is an arc, indicating a transition from the angular position \(Re^{i\frac{\pi}{4}}\) to \(R\) along a radial path. 3. **Arrows**: - The arrows on the graph indicate the direction of the contour. It moves from the origin along a straight line to \(Re^{i\frac{\pi}{4}}\) and then follows an arc back to \(R\). This graph is typically used in the context of complex analysis to illustrate contour integration, a crucial concept for evaluating integrals over complex functions using paths in the complex plane.
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Transcribed Image Text:The image above displays a graph depicting the path of a contour in the complex plane. The contour starts at the origin (0, 0), moves linearly to the point represented by \(Re^{i\frac{\pi}{4}}\), and then follows a curved path back to the point \(R\) on the real axis. ### Graph Explanation: 1. **Axes**: - The horizontal axis is labeled \(R\) and represents the real part of the complex plane. - The vertical axis is not labeled but typically represents the imaginary part of the complex plane. 2. **Path Description**: - The contour begins at the origin, indicated by the point \(0\) at the intersection of the two axes. - It then proceeds linearly to the point labeled \(Re^{i\frac{\pi}{4}}\). This point is at a distance \(R\) from the origin and makes an angle of \( \frac{\pi}{4}\) (or 45 degrees) with the positive real axis. - The path then follows a curved line back to the point \(R\) on the real axis. This curved part is an arc, indicating a transition from the angular position \(Re^{i\frac{\pi}{4}}\) to \(R\) along a radial path. 3. **Arrows**: - The arrows on the graph indicate the direction of the contour. It moves from the origin along a straight line to \(Re^{i\frac{\pi}{4}}\) and then follows an arc back to \(R\). This graph is typically used in the context of complex analysis to illustrate contour integration, a crucial concept for evaluating integrals over complex functions using paths in the complex plane.
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