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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 a Complex Number

Find the absolute value \(|z|\) and the argument \(\arg(z)\) of \(z = 1 + i\). Express it in polar form \(z = re^{i\theta}\).

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

Using Euler's formula, verify that \(e^{i\pi} + 1 = 0\). List the five mathematical constants (\(e, i, \pi, 1, 0\)) contained in this expression.

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B-3. Product in Polar Form

Given the complex numbers \(z_1 = 2e^{i\pi/3}\) and \(z_2 = 3e^{i\pi/6}\), find the product \(z_1 z_2\) in polar form, and then convert it to rectangular form \(a + bi\).

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

Verify that \(f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + 2ixy\) satisfies the Cauchy-Riemann relations.

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

Verify that \(f(z) = |z|^2 = x^2 + y^2\) does not satisfy the Cauchy-Riemann equations.

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

Using the definition of the Wirtinger derivative

\[ \partial_z = \frac{1}{2}(\partial_x - i\partial_y) \]

show that \(\partial_z(z^2) = 2z\) for \(f(z) = z^2\) (where \(z = x + iy\)).

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

Find the residue of \(f(z) = 1/(z-1)\) at \(z = 1\).

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

Write the Laurent expansion of \(f(z) = 1/z^2\) about \(z = 0\), and confirm that the residue is \(0\).

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

Write the Laurent expansion of \(e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n! z^n}\) at \(z = 0\), and confirm that the residue (the coefficient of \(z^{-1}\)) is \(1\). What type of singularity is \(z = 0\)?

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Medium

M-1. Residue Theorem: Two Poles

Evaluate \(\oint_{|z|=2} \frac{dz}{z(z-1)}\) using the residue theorem. (There are two poles at \(z = 0\) and \(z = 1\) inside the circle \(|z| = 2\).)

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

Find the residues of \(f(z) = \frac{z}{(z-1)(z-2)}\) at \(z = 1\) and \(z = 2\), respectively, and compute \(\oint_{|z|=3} f(z) dz\).

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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 \(w = 1/z\)

Determine where the conformal mapping \(w = 1/z\) maps the unit circle \(|z| = 1\) in the \(z\)-plane to in the \(w\)-plane. Also, determine where it maps the real axis in the \(z\)-plane to in the \(w\)-plane.

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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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