DescriptionSince the Fourier coefficients of an automorphic form along the nilpotent radical of parabolic subgroup are expressed in terms of Whittaker functions, a better understanding of their growth in every direction would be useful in the study of automorphic forms. Bump and Huntley (1995) used an integral formula which was found by Vinogradov, Takhtadzhyan (1978), and Stade (1988) to obtain precise information of the spherical Whittaker functions M(ν1,ν2) (y1, y2) as both y1 and y2 → ∞. To (1995) used a method similar to the characteristic method in the theory of differential equations to compute the leading exponents of asymptotic expansions of a basis of Whittaker functions in the positive Weyl chamber for a split semi-simple Lie group over R,, which, in particular, yields a solution to Zuckerman's conjecture for SL(3, R). Templier (2015) has recently used an integral representation by Givental to show To's result: the exponential growth of M(ν1,ν2) (y1, y2) for y1, y2 ≥ 1 as either or both y1, y2 → ∞. In this thesis I use a new formula which was derived by Ishii and Stade (2007) to obtain the asymptotic expansions of M(ν1,ν2) (t,1/tp) and M(ν1,ν2)(1/tp, t) as t → ∞ where 2 ≤ p ∈ 1/2Z then successfully prove an analog of the Multiplicity One Theorem in these directions, namely that in certain circumstances the moderate growth condition in the theory of automorphic forms is automatic.