I know that the general solution to temperature distribution on a ring of circumference length 1 is given by the Fourier series
$$T(x,t) = \sum_{n= - \infty}^{n=\infty} c_n(0) e^{-2 \pi^2 n^2 t} e^{2 \pi i n x}$$
where
$$c_n(0) = \int_0^1 T(x,0) e^{-2 \pi i n x} dx$$
If I start with a temperature distribution
$$T(x,0) = cos(2 \pi x) + 1$$
I calculate $c_{n \ne 0}(0) = 0$ and $c_{n=0}(0)=1$.
Consequently
$$T(x,t) = 1$$
which contradicts the original assumption that $T(x,0) = cos(2 \pi x)+1$.
Can anyone tell me where I'm going wrong?
Thank you.