Magic angles

The function ${u(x, y, z) = 2z^2 - x^2 - y^2}$ is notable for the following combination of properties:

1. It is a harmonic function: ${\Delta u = -2 - 2 + 4 = 0}$
2. It vanishes at the origin ${(0, 0, 0)}$ together with its gradient.
3. It is positive on the cone ${C=\{(x, y, z) : z > \sqrt{(x^2+y^2)/2}\}}$

The cone C has the opening angle ${\theta_3 = \cos^{-1}(1/\sqrt{3}) \approx 57.4^\circ}$ which is known as the magic angle in the context of NMR spectroscopy. Let us consider the mathematical side of its magic.

If C is replaced by any larger cone, the properties 1-2-3 cannot be satisfied by a harmonic function in a neighborhood of the origin. That is, C is the largest cone such that a harmonic function can have a critical point at its vertex which is also a point of its extremum on the cone. Why is that?

Let ${u}$ be a harmonic function in some neighborhood of ${(0,0,0)}$ and suppose ${u(0,0,0)=0}$, ${\nabla u(0,0,0) = 0}$, and ${u>0}$ on some cone ${C = \{(x, y, z) \colon z>c \sqrt{x^2+y^2+z^2} \}}$. Expand ${u}$ into a sum of polynomials ${p_2+p_3+\cdots }$ where each ${p_k}$ is a harmonic homogeneous polynomial of degree ${k}$. Let ${m}$ be the smallest integer such that ${p_m}$ is not identically zero. Then ${p_m}$ has the same properties as ${u}$ itself, since it dominates the other terms near the origin. We may as well replace ${u}$ by ${p_m}$: that is, ${u}$ is a spherical harmonic from now on.

Rotating ${u}$ around the ${z}$-axis preserves all the properties of interest: harmonic, positive on the cone, zero gradient at the origin. Averaging over all these rotations we get a rotationally symmetric function known as a zonal spherical harmonic. Up to a constant factor, such a function is given by ${P_m(\cos \phi)}$ where ${\phi}$ is a spherical coordinate (angle with the ${z}$-axis) and ${P_m}$ is the Legendre polynomial of degree ${m}$.

The positivity condition requires ${P_m(t) > 0}$ for ${t>c}$. In other words, the bound on ${\theta}$ comes from the greatest zero of the Legendre polynomial. As is true for orthogonal polynomials in general, the zeros are interlaced: that is, the zeros of ${P_m=0}$ appear strictly between any two consecutive zeros of ${P_{m+1}}$. It follows that the value of the greatest zero grows with ${m}$. Thus, it is smallest when ${m=2}$. Since ${P_2(t) = (3t^2-1)/2}$, the zero of interest is ${1/\sqrt{3}}$, and we conclude that ${c \ge 1/\sqrt{3}}$. Hence the magic angle.

The magic angle is easy to visualize: it is formed by an edge of a cube and its space diagonal. So, the magic cone with vertex at (0,0,0) is the one that passes through (1, 1, 1), as shown above.

In other dimensions the zonal harmonics are expressed in terms of Gegenbauer polynomials (which reduce to Chebyshev polynomials in dimensions 2 and 4). The above argument applies to them just as well. The relevant Gegenbauer polynomial of degree ${2}$ is ${nx^2-1}$ up to a constant. Thus, in ${\mathbb R^n}$ the magic angle is ${\cos^{-1}(1/\sqrt{n})}$, illustrated by the harmonic function ${u(x)=nx_n^2 - |x|^2}$.

This analysis is reminiscent of the Hopf Lemma which asserts that if a positive harmonic function in a smooth domain has boundary value 0 at some point, the normal derivative cannot vanish at that point. The smoothness requirement is typically expressed as the interior ball condition: one can touch every boundary point by a ball contained in the domain. The consideration of magic angles shows that if the function is also harmonic in a larger domain, the interior ball condition can be replaced by the interior cone condition, provided that the opening angle of the cone is greater than ${\cos^{-1}(1/\sqrt{n})}$.

Misuse of bi-

The prefix bi- is apt to cause confusion; how often does a bimonthly event occur? Its usage in mathematics is not free of inconsistency, either. Compare:

• Biholomorphic: a holomorphic map $f$ such that $f^{-1}$ is also holomorphic
• Biharmonic: a function $u$ such that $\Delta \Delta u=0$

Both of these usages are well established. Let’s switch them around just for the fun of it.

(1) Recall that $f\colon \Omega\to\mathbb C$ (where $\Omega\subset \mathbb C$) is holomorphic if $\displaystyle\frac{\partial }{\partial \bar z}f=0$. Following the second usage pattern, we would call $f$ “biholomorphic” if $\displaystyle \frac{\partial}{\partial \bar z}\frac{\partial}{\partial \bar z} f=0$. What can we say about such functions? Let $\displaystyle g=\frac{\partial }{\partial \bar z}f$. Clearly, $g$ is holomorphic. Now introduce $h(z)=f(z)-\bar zg(z)$. This function satisfies $\displaystyle \frac{\partial }{\partial \bar z}h(z) = g(z)-g(z)=0$, i.e., it is also holomorphic. Conclusion: the solutions of the equation $\displaystyle \left(\frac{\partial}{\partial \bar z}\right)^2 f=0$ are precisely the functions of the form $f(z)=h(z)+\bar z g(z)$ where $h$ and $g$ are holomorphic.

This representation formula tells us a lot about “biholomorphic” functions. They are very smooth: the real and imaginary parts are real-analytic. They are locally invertible outside of the set $\{z\colon |h'(z)+\bar zg'(z)|=|g(z)|\}$, which is usually 1-dimensional (compare to the discreteness of the branch set of holomorphic functions). They are not open in general: consider $f(z)=z\bar z = |z|^2$ for example. The modulus $|f|$ does not satisfy the maximum principle: consider $f(z)=1-|z|^2$. But Liouville’s theorem does hold: if $f$ is bounded in $\mathbb C$, it is constant.

(2) A map $f\colon \Omega\to\mathbb C$ (where $\Omega\subset \mathbb C$) is harmonic if $\Delta f=0$, or, put another way, if $f=u+iv$ where $u$ and $v$ are real harmonic functions. Following the first usage pattern, we would call $f$ “biharmonic” if $f^{-1}$ exists and is also harmonic. Any invertible holomorphic map clearly satisfies this definition. Anything else? Well, there are affine maps $f(z)=a z+b\bar z+c$ which are invertible as long as $|a|\ne |b|$ (here $a,b,c\in\mathbb C$ are constant.) The inverse of an affine map is also affine, hence harmonic. Anything else? Nothing comes to mind.

But in 1945 Gustave Choquet gave this example:

$\displaystyle x+iy\mapsto x+iv,\qquad \tan v \tan y = \tanh x$

That’s not a typo: both the regular tangent and its hyperbolic cousin appear in the formula. It is not at all obvious (but true) that $v$ is a harmonic function of $x$ and $y$. It is rather obvious that the map is an involution, hence “biharmonic” according to our definition.

Now that we have one “biharmonic” map that’s neither holomorphic nor affine, are there others? No (apart from trivial variations of $f$). Choquet attributed this uniqueness result to Jacques Deny who did not publish it. The first published classification of “biharmonic” maps appeared in a 1987 paper by Edgar Reich in a rather different form:

If $f$ and $f^{-1}$ are harmonic, then $f$ is either holomorphic, affine, or of the form $\displaystyle f(z)=\alpha(\beta z+2i\arg (\gamma-e^{\beta z}))+\delta$ where $\alpha,\beta,\gamma,\delta$ are complex constants such that $\alpha\beta\gamma\ne 0$ and $|e^{-\beta z}|<\gamma$ in the domain of $f$.

The proof can be found in Peter Duren’s book from which I quoted the statement of this result.