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Appendix E Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Absolute Value and Argument of Complex Numbers

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B-2. Euler's Formula \(e^{i\pi}+1=0\)

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B-3. Product in Polar Form

Regular Functions (E.3-E.4)

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B-4. Cauchy-Riemann: Verification with \(z^2\)

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B-5. Cauchy-Riemann: \(|z|^2\) Fails

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B-6. \(\partial_z(z^2) = 2z\)

Laurent Series and Residues (E.5-E.6)

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B-7. Residue of \(1/(z-1)\)

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B-8. Laurent Expansion and Residue of \(1/z^2\)

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B-9. Laurent Expansion of \(e^{1/z}\)

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Medium

M-1. Residue Theorem: Two Poles

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M-2. Residues of \(z/[(z-1)(z-2)]\)

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M-3. Composition of Möbius Transformations

Given two Möbius transformations \(w_1(z) = (z+1)/(z-1)\) and \(w_2(w) = 2w + 3\), find the composed transformation \(w_2(w_1(z))\) and verify that the corresponding matrix product is consistent.

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Advanced

A-1. Conformal Mapping \(w = 1/z\)

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A-2. Cross Terms of \(\partial X\) and \(\bar\partial X\)

In the main text E.8 "Green's Function for the 2D Free Field", the two-point function \(\langle X(z,\bar z)\, X(w,\bar w)\rangle = -\frac{\alpha'}{2}\ln\lvert z-w\rvert^2\) was derived. From this result, show the following:

(a) \(\langle \partial X(z)\, \bar\partial X(w)\rangle\) vanishes for \(z \neq w\). (b) \(\langle \bar\partial X(\bar z)\, \bar\partial X(\bar w)\rangle = -\frac{\alpha'}{2}\, \frac{1}{(\bar z - \bar w)^2}\) (the anti-holomorphic OPE).

Hint: Decompose \(\ln\lvert z-w\rvert^2 = \ln(z-w) + \ln(\bar z - \bar w)\), and use the fact that \(\partial_z\) acts only on the holomorphic part while \(\partial_{\bar z}\) acts only on the anti-holomorphic part.

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