Consider the heat equation u t
​
= 9
1
​
[u φφ
​
+(cotφ)u φ
​
] on the surface of the sphere of radius 3 , with the initial condition u(φ,0)=1−3cos(2φ) the solution is u(φ,t)=∑ n=0
[infinity]
​
a n
​
P n
​
(cosφ)e −n(n+1)t/9
, P n
​
(x) are Legendre polynomials. Find the coefficient a 0
​
a) 3 b) −1 c) 4 d) 1 e) 2 f) 0