2л = 1/2 √² 2л = 1/2 √² 2π T/F R²-z. Z (Reiy–z) (Re-ių – 7.) (Re) dys (Rey+z) (Rei −7)+(Reių −z) (Re-iy + 7) (Reiy–z) (Re-iy — 7.) f (Rew) dys

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Bartleby sent this a part of a solution yesterday, and I don't see how they are getting from the first line to the second.  Is some identity being used?  I just can't seem to follow the work at this point.  Thanks.

The mathematical content shown above features an expression involving integrals, complex variables, and functions, suitable for inclusion in an educational website that covers advanced topics in mathematics, specifically in the area of complex analysis.

The expression can be broken down as follows:

First line:
\[ 
= \frac{1}{2\pi} \int_0^{2\pi} \left[ \frac{R^2 - z \bar{z}}{(Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})} \right] f \left( Re^{i\psi} \right) d\psi 
\]

Second line:
\[ 
= \frac{1}{2\pi} \int_0^{2\pi} \left[ \frac{ (Re^{i\psi} + z)(Re^{i\psi} - \bar{z}) + (Re^{i\psi} - z)(Re^{-i\psi} + \bar{z}) }{(Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})} \right] f \left( Re^{i\psi} \right) d\psi 
\]

### Explanation:

1. **First Integral:**
   - The first line represents an integral over the interval from \(0\) to \(2\pi\).
   - The integral's kernel contains a fraction, where the numerator is \(R^2 - z \bar{z}\).
   - The denominator is the product \((Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})\), with \(e^{i\psi}\) and \(e^{-i\psi}\) representing complex exponentials, thus hinting at the use of polar coordinates in the complex plane.
   - The function \(f(Re^{i\psi})\) is being integrated with respect to \(d\psi\).

2. **Second Integral:**
   - The second line rearranges the above expression.
   - The new integral's kernel has a refined and expanded form in the numerator as \((Re^{i\psi} + z)(Re^{i\psi} - \bar{z}) + (Re^{i\psi} - z)(Re^{-i\psi} + \bar{z})\), which simplifies the initial fraction.
Transcribed Image Text:The mathematical content shown above features an expression involving integrals, complex variables, and functions, suitable for inclusion in an educational website that covers advanced topics in mathematics, specifically in the area of complex analysis. The expression can be broken down as follows: First line: \[ = \frac{1}{2\pi} \int_0^{2\pi} \left[ \frac{R^2 - z \bar{z}}{(Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})} \right] f \left( Re^{i\psi} \right) d\psi \] Second line: \[ = \frac{1}{2\pi} \int_0^{2\pi} \left[ \frac{ (Re^{i\psi} + z)(Re^{i\psi} - \bar{z}) + (Re^{i\psi} - z)(Re^{-i\psi} + \bar{z}) }{(Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})} \right] f \left( Re^{i\psi} \right) d\psi \] ### Explanation: 1. **First Integral:** - The first line represents an integral over the interval from \(0\) to \(2\pi\). - The integral's kernel contains a fraction, where the numerator is \(R^2 - z \bar{z}\). - The denominator is the product \((Re^{i\psi} - z)(Re^{-i\psi} - \bar{z})\), with \(e^{i\psi}\) and \(e^{-i\psi}\) representing complex exponentials, thus hinting at the use of polar coordinates in the complex plane. - The function \(f(Re^{i\psi})\) is being integrated with respect to \(d\psi\). 2. **Second Integral:** - The second line rearranges the above expression. - The new integral's kernel has a refined and expanded form in the numerator as \((Re^{i\psi} + z)(Re^{i\psi} - \bar{z}) + (Re^{i\psi} - z)(Re^{-i\psi} + \bar{z})\), which simplifies the initial fraction.
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