Abstract
Fourier analysis is perhaps the most important single tool in the study of linear partial differential equations. It serves in several ways, the most basic–and historically the first–being to give specific formulas for solutions to various linear PDE with constant coefficients, particularly the three classics, the Laplace, wave, and heat equations:
with Δ = ∂2∕∂x12 + ⋯ + ∂2∕∂xn2. The Fourier transform accomplishes this by transforming the operation of ∂∕∂xj to the algebraic operation of multiplication by iξj. Thus the (0.1) are transformed to algebraic equations and to ODE with parameters.
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L. Bers, F. John, and M. Schechter, Partial Differential Equations, Wiley, New York, 1964.
R. Churchill, Fourier Series and Boundary Value Problems, McGraw Hill, New York, 1963.
J. Dadok and M. Taylor, Local solvability of constant coefficient linear PDE as a small divisor problem, Proc. AMS 82(1981), 58–60.
W. Donoghue, Distributions and Fourier Transforms, Academic, New York, 1969.
A. Erdelyi, Asymptotic Expansions, Dover, New York, 1965.
A. Erdelyi et al., Higher Transcendental Functions, Vols. I–III (Bateman Manuscript Project), McGraw-Hill, New York, 1953.
G. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, N. J., 1976.
F. G. Friedlander, Sound Pulses, Cambridge University Press, Cambridge, 1958.
A. Friedman, Generalized Functions and Partial Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1963.
I. M. Gelfand and I. Shilov, Generalized Functions, Vols. 1–3, Fizmatgiz, Moscow, 1958.
R. Goldberg, Fourier Transforms, Cambridge University Press, Cambridge, 1962.
E. Hille and R. Phillips, Functional Analysis and Semi-groups, Colloq. Publ. AMS, Providence, R.I., 1957.
E. Hobson, The Theory of Spherical Harmonics, Chelsea, New York, 1965.
L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 1–2, Springer, New York, 1983.
L. K. Hua, Introduction to Number Theory, Springer, New York, 1982.
F. John, Partial Differential Equations, Springer, New York, 1975.
Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York, 1976.
O. Kellogg, Foundations of Potential Theory, Dover, New York, 1954.
T. Körner, Fourier Analysis, Cambridge University Press, Cambridge, 1988.
E. Landau, Elementary Number Theory, Chelsea, New York, 1958.
N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.
J. Leray, Hyperbolic Differential Equations, Princeton University Press, Princeton, N. J., 1952.
S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973.
F. Olver, Asymptotics and Special Functions, Academic, New York, 1974.
S. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1988.
M. Pinsky and M. Taylor, Pointwise Fourier inversion: a wave equation approach, J. Fourier Anal. 3 (1997), 647–703.
W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes; the Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
H. Rademacher, Lectures on Elementary Number Theory, Chelsea, New York, 1958.
M. Reed and B. Simon, Methods of Mathematical Physics, Academic, New York, Vols. 1,2, 1975; Vols. 3,4, 1978.
F. Riesz and B. S. Nagy, Functional Analysis, Ungar, New York, 1955.
W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1976.
L. Schwartz, Théorie des Distributions, Hermann, Paris, 1950.
I. Stakgold, Boundary Value Problems of Mathematical Physics, Macmillan, New York, 1968.
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N. J., 1970.
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press, Princeton, N. J., 1971.
M. Taylor, Noncommutative Harmonic Analysis, Math. Surveys and Monographs, No. 22, AMS, Providence, 1986.
F. Treves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, New York, 1966.
G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944.
E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927.
N. Wiener, Tauberian theorems, Ann. Math. 33(1932), 1–100.
K. Yosida, Functional Analysis, Springer, New York, 1965.
A. Zygmund, Trigonometrical Series, Cambridge University Press, Cambridge, 1969.
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A The mighty Gaussian and the sublime gamma function
A The mighty Gaussian and the sublime gamma function
The Gaussian function \({e}^{-\vert x{\vert }^{2} }\) on \({\mathbb{R}}^{n}\) is an object whose study yields many wonderful identities. We will use the identity
which was established in (3.18), to compute the area An − 1 of the unit sphere Sn − 1 in \({\mathbb{R}}^{n}\). This computation will bring in Euler’s gamma function, and other results will flow from this. Switching to polar coordinates for the right side of (A.1), we have
where the gamma function is defined by
for Re z > 0. Thus we have the formula
To be satisfied with this, we need an explicit evaluation of Γ(n∕2). This can be obtained from Γ(1∕2) and Γ(1) via the following identity:
for Re z > 0, where we used integration by parts. The definition (A.3) clearly gives
Thus, for any integer k ≥ 1,
Note that, for n = 2, we have A1 = 2π∕Γ(1), so (A.6) agrees with the fact that the circumference of the unit circle is 2π (which, of course, figured into the proof of (3.18), via (3.20). In case n = 1, we have A0 = 2, which by (A.4) is equal to 2π1∕2∕Γ(1∕2), so
Again using (A.5), we see that, when k ≥ 1 is an integer,
In particular, Γ(3∕2) = (1∕2)Γ(1∕2) = π1∕2∕2, so A2 = 2π3∕2∕(π1∕2∕2) = 4π,which agrees with the well known formula for the area of the unit sphere in \({\mathbb{R}}^{3}\).
Note that while Γ(z) defined by (A.3) is a priori holomorphic for Re z positive, the equation (A.5) shows that Γ(z) has a meromorphic extension to the entire complex plane, with simple poles at \(z = 0,-1,-2,\ldots\,\). It turns out that Γ(z) has no zeros, so 1∕Γ(z) is an entire analytic function. This is a consequence of the identity
which we now establish. From (A.4) we have (for 0 < Re z < 1)
where we have used the change of variables u = s + t, v = t∕s. With v = ex, the last integral is
which is holomorphic for 0 < Re z < 1, and we want to show that it is equal to the right side of (A.10) on this strip. It suffices to prove identity on the line \(z = 1/2 + i\xi, \xi \in\mathbb{R}\); then (A.12) is equal to the Fourier integral
To evaluate this, shift the contour of integration from the real line to the line Im x = −2π. There is a pole of the integrand at x = − πi, and we have (A.13) equal to
Consequently, (A.13) is equal to
and since π/ sinπ(1∕2 + iξ) = π∕coshπξ, the demonstration of (A.10) is complete.
The integral (A.3) and also the last integral in (A.11) are special cases of the Mellin transform:
If we evaluate this on the imaginary axis:
given appropriate growth restrictions on f, this is related to the Fourier transform by a change of variable:
The Fourier inversion formula and Plancherel formula imply
and
In some cases, as seen above, one evaluates ℳf(z) on a vertical line other than the imaginary axis, which introduces only a slight wrinkle.
An important identity for the gamma function follows from taking the Mellin transform with respect to y of both sides of the subordination identity
(if y > 0, A > 0), established in §5; see (5.22). The Mellin transform of the left side is clearly Γ(z)A− z. The Mellin transform of the right side is a double integral, which is readily converted to a product of two integrals, each defining gamma functions. After a few changes of variables, there results the identity
known as the duplication formula for the gamma function. In view of the uniqueness of Mellin transforms, following from (A.18) and (A.19), the identity (A.22) conversely implies (A.21). In fact, (A.22) was obtained first (by Legendre) and this argument produces one of the standard proofs of the subordination identity (A.21).
There is one further identity, which, together with (A.5), (A.10), and (A.22), completes the list of the basic elementary identities for the gamma function. Namely, if the beta function is defined by
(with u = s∕(1 − s)), then
To prove this, note that since
we have
so
as asserted.
The four basic identities proved above are the workhorses for most applications involving gamma functions, but fundamental insight is provided by the identities (A.27) and (A.31) below. First, since 0 ≤ e− t − (1 − t∕n)n ≤ e− t ⋅t2∕n for 0 ≤ t ≤ n, we have, for Re z > 0,
Repeatedly integrating by parts gives
which yields the following result of Euler:
Using the identity (A.5), analytically continuing Γ(z), we have (A.27) for all z, other than \(0,-1,-2,\ldots \). We can rewrite (A.27) as
If we denote by γ Euler’s constant:
then (A.28) is equivalent to
that is, to the Euler product expansion
It follows that the entire analytic function 1∕Γ(z)Γ(− z) has the product expansion
Since Γ(1 − z) = − zΓ(− z), by virtue of (A.10) this last identity is equivalent to the Euler product expansion
It is quite easy to deduce the formula (A.5) from the Euler product expansion (A.31). Also, to deduce the duplication formula (A.22) from the Euler product formula is a fairly straightforward exercise.
Finally, we derive Stirling’s formula, for the asymptotic behavior of Γ(z) as z → + ∞. The approach uses the Laplace asymptotic method, which has many other applications. We begin by setting t = sz and then s = ey in the integral formula (A.3), obtaining
The last integral is of the form
where φ(y) = ey − y has a nondegenerate minimum at y = 0; φ(0) = 1, φ′(0) = 0, φ′(0) = 1. If we write 1=A(y) + B(y), A ∈ C0∞((−2,2)), A(y) = 1 for |y| ≤ 1, then the integral (A.35) is readily seen to be
We can make a smooth change of variable x = ξ(y) such that ξ(y) = y + O(y2), φ(y) = 1 + x2∕2, and the integral in (A.36) becomes
where \({A}_{1}\in{C}_{0}^{\infty }(\mathbb{R}),\ {A}_{1}(0) = 1\), and it is easy to see that, as z → + ∞,
In fact, if z = 1∕2t, then (A.38) is equal to (4πt)1∕2 u(t,0), where u(t, x) solves the heat equation, ut − uxx = 0, u(0, x) = A1(x). Returning to (A.34), we have Stirling’s formula:
Since n!=Γ(n + 1), we have in particular that
as n → ∞.
Regarding this approach to the Laplace asymptotic method, compare the derivation of the stationary phase method in Appendix B of Chap. 6.
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Taylor, M.E. (2011). Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE. In: Partial Differential Equations I. Applied Mathematical Sciences, vol 115. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7055-8_3
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