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How were the real spherical harmonics derived?

The complex spherical harmonics:

$$ Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi} $$

But the "real" spherical harmonics are given on this wiki page as

$$ Y_{lm} = \begin{cases} \frac{1}{\sqrt{2}} ( Y_l^m + (-1)^mY_l^{-m} ) & \text{if } m > 0 \\ Y_l^m & \text{if } m = 0 \\ \frac{1}{i \sqrt{2}}( Y_l^{-m} - (-1)^mY_l^m) & \text{if } m < 0 \end{cases} $$

  • Note: $Y_{lm} $ is the real spherical harmonic function and $Y_l^m$ is the complex-valued version (defined above)

What's going on here? Why are the real spherical harmonics defined this way and not simply as $ \Re{( Y_l^m )} $ ?

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The page actually suggests the answer when it says "The harmonics with $m > 0$ are said to be of cosine type, and those with $m < 0$ of sine type." Recall how one switches between the complex exponential functions $\{e^{imx}\colon m\in \mathbb Z\}$ and the trigonometric functions: it's done with the formulas $$\cos mx=\frac{e^{imx}+e^{-imx}}{2}$$ and $$\sin mx=\frac{e^{imx}-e^{-imx}}{2i}$$ Taking only real parts would not give you the sines.

Since $\cos (-mx)=\cos mx$ and $\sin(-mx)=-\sin mx$, we don't need all values of $m$ in both families. We can remove the redundant functions and enumerate the entire trigonometric basis by $m\in\mathbb Z$ as follows: $\{\cos mx\colon m\ge 0\}\cup \{\sin mx\colon m<0\}$. This is essentially what the wiki page does.

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  • $\begingroup$ A good answer. But is there a real reason for doing this? Is there an underlying reason why they made them "cosine type" and "sine type"? $\endgroup$ – bobobobo May 14 '12 at 19:43
  • $\begingroup$ For one thing, it makes sense to keep even and odd basis functions (w.r.t. $\varphi$) separately; for instance if you want to expand an even function in this basis, you only need the even basis functions. But really: if you have a basis for $L^2_{\mathbb C}$ which involves complex exponential functions, and you want a basis for $L^2_{\mathbb R}$, what are you going to do? Of course, you will rewrite exponentials in terms of cosines and sines. $\endgroup$ – user31373 May 14 '12 at 20:27
  • $\begingroup$ Does the Addition Theorem still holds for Real spherical harmonics ? $\endgroup$ – user52342 Mar 6 at 10:38
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As far as I remember, real Spherical Harmonics are real functions that still have an eigenvalue respect to L operator (angular moment). They are no longer eigenfunctions of L_z operator (M value, as they mix m and -m). On another hand, real Spherical Harmonics are also orthonormal.

Real part of Spherical Harmonics are not eigenfunctions of L operator, nor orthonormal.

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Why are the real spherical harmonics defined this way and not simply as $\Re{(Y_l^m)}$?

Well yes it is! The real spherical harmonics can be rewritten as followed: $$Y_{lm} = \begin{cases} \sqrt{2}\Re{(Y_l^m)}=\sqrt{2}N_l^m\cos{(m\phi)}P_l^m(\cos \theta) & \text{if } m > 0 \\ Y_l^0=N_l^0P_l^0(\cos \theta) & \text{if } m = 0 \\ \sqrt{2}\Im{(Y_l^m)}=\sqrt{2}N_l^{|m|}\sin{(|m|\phi)}P_l^{|m|}(\cos \theta) & \text{if } m < 0 \end{cases} $$

(Some texts denote lowercase $y$ for real harmonics). If you look at the table, the negative $m$ is the imaginary part of the positive $m$ (but not vice versa).

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