下面两个软件计算的结果完全不同！

ylm      Normalized real spherical harmonics (Dahlen & Tromp B.72) 

 Calculates unit-normalized real spherical harmonics, 计算单位归一化实球谐函数

sphere-lab Hilbert Space Methods in Signal Processing

总感觉计算结果不太对，比如：

ylm(60,20,pi/2-20*pi/180,97*pi/180) = 0.1587

sphHarm(60,20,pi/2-20*pi/180,97*pi/180) =  -0.1338 + 0.1123i

real_sphHarm(60,20,pi/2-20*pi/180,97*pi/180) =  -0.1892

究竟球谐函数Ylm到底是多少？实球谐函数和复球谐函数的关系？

球谐（Spherical Harmonics，SH）函数有实数形式和复数形式，SH函数一般是在虚数上定义的。

实球谐函数

实球谐函数推导

实际公式和代码对应起来，应该是sphere-lab 结果更为可靠！

function Ylm = real_sphHarm(l, m, lTheta, lPhi)

% Ylm = sphHarm(l, m, lTheta, lPhi)

% Create a real spherical harmonic function of degree 'l' and order 'm'. See

% README.txt for more details.

%   Author: chenchao.

%   Created: Mar 18, 2022.

if (mod(m, 2) == 1)

  neg1_m = -1;

else

  neg1_m = 1;

end

% TODO: We currently calculate Y_{lm} by definition, but it could be faster.

if (m > 0)

  Ylm = 1/sqrt(2)*(sphHarm(l, -m, lTheta, lPhi) + sphHarm(l,  m, lTheta, lPhi) * neg1_m);

elseif (m == 0)

  Ylm = sphHarm(l, 0, lTheta, lPhi);

else

  Ylm = 1i/sqrt(2)*(sphHarm(l,  m, lTheta, lPhi) - sphHarm(l, -m, lTheta, lPhi) * neg1_m);

end

Ylm = real(Ylm);

实球谐函数