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In Cartesian comoving coordinates and signature $(-,+,+,+)$, a spatially flat FLRW universe has

If the present radiation and cosmological-constant densities are respectively $\beta$ and $1$ times the present critical density, while $a_0=1$, positive curvature givesTurning points are therefore the positive roots of $x^2-\beta x+\beta$ with $x=a^2$.

For energy density $\rho$, pressure $P$, and cosmological constant $\Lambda$,

Combining the perfect-fluid continuity equation withshows thatis constant in time. This constant is the spatial-curvature integration constant in the first Friedmann equation.

A self-similar object is assembled from smaller copies of itself, possibly after scaling or other transformations.

With the standard Fourier convention, Plancherel theorem applied to $1_{[-1,1]}$ gives

With one common sign convention,Reversing the denominator reverses the sign.

As a tempered distribution on ${\mathbb{R}}^3$,for a nonzero convention-dependent constant $C$. One proof writesand applies the Fourier transform of a Gaussian before substituting $u=|\xi {|}^2/(4t)$.

If $f_{\lambda }(x)=f(\lambda x)$ on ${\mathbb{R}}^n$ with $\lambda >0$, thenand consequently

On $(-\pi ,\pi )$,Parseval's identity and direct integration of the square give

A Fourier cosine series has the formIt represents the even function obtained by reflecting its data across the origin.

For an integrable function of period $L>0$, its complex Fourier coefficients measure its inner product with each complex exponential mode:For a real function of period $2\pi$, the real coefficient convention of Fourier series gives ${\hat{f}}_0=a_0/2$ and ${\hat{f}}_n=(a_n-i{b}_n)/2$ for $n\ge 1$, with ${\hat{f}}_{-n}=\overline{{\hat{f}}_n}$. For $f(x)=1/2+\cos x+\frac{1}{2}\sin (2x)$, the only nonzero complex Fourier coefficients are ${\hat{f}}_0=1/2$, ${\hat{f}}_{\pm 1}=1/2$, ${\hat{f}}_2=-i/4$, and ${\hat{f}}_{-2}=i/4$. This explains why the real and complex lists look different while representing the same function.

A trigonometric polynomial is a finite linear combination of complex exponentials $e^{inx}$, or equivalently of sines and cosines. If it vanishes on an interval, all its coefficients vanish.

The Leibniz formula is the conditionally convergent alternating series

Fourier restriction theory studies estimates for Fourier transforms restricted to curved submanifolds and for the corresponding extension operators.

For independent Rademacher signs ${\epsilon }_n$, every $0<p<\infty$ admits constants $A_p,B_p>0$ such that

With $\hat{f}(k)={\int }_{\mathbb{R}}f(x)e^{-ikx}dx$ and $a>0$,

For $f,g\in L^2({\mathbb{R}}^n)$ and the convention $\hat{f}(\xi )=\int f(x)e^{-ix\cdot \xi }dx$,

For the unnormalized angular-frequency transform,After normalization, the Fourier transform extends unitarily to $L^2$. For integrable $f$ and $\hat{f}$, inversion and Fubini give the identity directly.

If $f\in L^1({\mathbb{R}}^n)$, then its Fourier transform is continuous and tends to zero as $|\xi |\to \infty$. Approximate $f$ in $L^1$ by a compactly supported smooth function; integration by parts makes the approximant's transform decay, while $\Vert \hat{f}-\hat{g}{\Vert }_{\infty }\le \Vert f-g{\Vert }_1$ transfers the conclusion.