Analytic Formulation of 21 cm Signal from Cosmic Dawn: Ly$\alpha$
  Fluctuations
  We present an analytic formalism to compute the fluctuating component of the
\ion{H}{1} signal and extend it to take into account the effects of partial
Lyman-$\alpha$ coupling during the era of cosmic dawn. We use excursion set
formalism to calculate the size distribution of randomly distributed
self-ionized regions. These ionization bubbles are surrounded by partially
heated and Lyman-$\alpha$ coupled regions, which create spin temperature $T_S$
fluctuations. We use the ratio of number of Lyman-$\alpha$ to ionizing photon
($f_L$) and number of X-ray photons emitted per stellar baryons ($N_{\rm
heat}$) as modeling parameters. Using our formalism, we compute the global
\ion{H}{1} signal, its auto-correlation and power spectrum in the redshift
range $10 \le z \le 30$ for the $\Lambda$CDM model. We check the validity of
this formalism for various limits and simplified cases. Our results agree
reasonably well with existing results from N-body simulations, in spite of
following a different approach and requiring orders of magnitude less
computation power and time. We further apply our formalism to study the
fluctuating component corresponding to the recent EDGES observation that shows
an unexpectedly deep absorption trough in global \ion{H}{1} signal in the
redshift range $15 <z< 19$. We show that, generically, the EDGES observation
predicts larger signal in this redshift range but smaller signal at higher
redshifts. We also explore the possibility of negative real-space
auto-correlation of spin temperature and show it can be achieved for partial
Lyman-$\alpha$ coupling in many cases corresponding to simplified models and
complete model without density perturbations.
