31B (Asymptotic Methods)

The intended integral representation of the modified Bessel function is

I_{0} (x) = \frac{1}{π} \int_{0}^{π} e^{x c o s θ} d θ .

I_{0}^{'} (x) = \frac{1}{π} \int_{0}^{π} cos θ e^{x c o s θ} d θ, I_{0}^{''} (x) = \frac{1}{π} \int_{0}^{π} cos^{2} θ e^{x c o s θ} d θ .

Since

\frac{d}{d θ} (sin θ e^{x c o s θ}) = (cos θ - x sin^{2} θ) e^{x c o s θ},

integration by parts and the vanishing endpoint term yield

x I_{0}^{''} + I_{0}^{'} - x I_{0} = 0

For large positive

x

, Laplace's method localizes the integral near the endpoint

θ = 0

. Put

θ = u / x

and use

x cos θ = x - \frac{u ^{2}}{2} + \frac{u ^{4}}{24 x} + O (x^{- 2} u^{6}) .

Extending the exponentially small tail to infinity and evaluating the resulting Gaussian integrals gives

I_{0} (x) = \frac{e ^{x}}{π x} (\int_{0}^{\infty} e^{- u^{2} /2} d u + \frac{1}{24 x} \int_{0}^{\infty} u^{4} e^{- u^{2} /2} d u + O (x^{- 2})),

and therefore

I_{0} (x) \sim \frac{e ^{x}}{2 π x} (1 + \frac{1}{8 x} + O (x^{- 2})) .

Now write

y = x^{- 1/2} w

. Direct differentiation gives

y^{'} = x^{- 1/2} w^{'} - \frac{1}{2} x^{- 3/2} w, y^{''} = x^{- 1/2} w^{''} - x^{- 3/2} w^{'} + \frac{3}{4} x^{- 5/2} w,

so substitution into the ordinary differential equation gives

w^{''} + (\frac{1}{4 x ^{2}} - 1) w = 0 .

Q (x) = 1 - 1/ (4 x^{2})

. The two approximate solutions are

w_{\pm} (x) \sim Q (x)^{- 1/4} exp (\pm \int^{x} Q (s) d s),

so at leading order

y_{\pm} (x) \sim x^{- 1/2} e^{\pm x}

. The growing solution agrees with the preceding asymptotic expansion, including its leading normalization

I_{0} (x) \sim e^{x} / 2 π x

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