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 Incoming links: Integer-order Bessel reflection identity

Bessel function of the first kind Created 2026-10-03 Updated 2026-11-05

J_{ν} (z) = \sum_{j = 0}^{\infty} \frac{( - 1 ) ^{j} ( z /2 ) ^{2 j + ν}}{j ! Γ ( j + ν + 1 )} .

The reciprocal of the gamma function is interpreted by its entire function continuation, which vanishes at

0, - 1, - 2, \dots

. For noninteger order, the factor

z^{ν}

requires a specified branch of a multivalued function. It need not be regular at zero: negative noninteger order can give a divergent leading power. For nonnegative integer order, the power series is regular and single valued, and negative integer orders are covered by the Integer-order Bessel reflection identity.^[ref]

For

Re ν > - 1/2

J_{ν} (z) = \frac{( z /2 ) ^{ν}}{Γ ( ν + 1/2 ) π} \int_{- 1}^{1} e^{i z t} (1 - t^{2})^{ν - 1/2} d t .

The endpoint condition on

ν

makes the weight integrable. The zero-order series begins

J_{0} (z) = 1 - z^{2} /4 + z^{4} /64 - \dots

, giving a concrete check of normalization and the origin behavior.

For a nonnegative integer

m

, expand the plane wave in powers of

ρ

e^{i ρ c o s θ} = \sum_{n = 0}^{\infty} \frac{( i ρ ) ^{n}}{n !} (\frac{e ^{i θ} + e ^{- i θ}}{2})^{n} .

The first contribution to the positive angular harmonic

e^{im θ}

occurs at

n = m

(i ρ)^{m} / (2^{m} m!)

. Comparing this with

a_{m} J_{m} (ρ) \sim a_{m} ρ^{m} / (2^{m} m!)

gives

a_{m} = i^{m}

for

m \geq 0

The plane wave is even in

θ

, so its

m

and

- m

a_{- m} J_{- m} = a_{m} J_{m}

, implying

a_{- m} = (- 1)^{m} i^{m} = i^{- m}

. Hence

a_{m} = i^{m} (m \in Z), e^{i ρ c o s θ} = m = - \infty \sum \infty i^{m} J_{m} (ρ) e^{im θ} .

This is the cosine form of the Jacobi-Anger expansion. The factorial asymptotic stated in the source is used for nonnegative order; applying it literally to negative integers would introduce an undefined factorial. The Integer-order Bessel reflection identity is the required extension, not a separate set of divergent radial solutions.