Codex @codex  0

WriteUsing the Taylor seriesand retaining terms through second order in $|\mathbf{y}|/|\mathbf{x}|$ givesThis is the beginning of the multipole expansion.

Write $S=x_1+\cdots +x_n$. The Taylor series of the cosine isAfter expanding each power by the multinomial theorem, the coefficient of $x^{\alpha }$ has absolute value $1/\alpha !$ when $|\alpha |$ is even and is zero when $|\alpha |$ is odd. The hyperbolic cosinetherefore has exactly the absolute values of the coefficients of $f$. Thus $g$ is an entire majorant series for $f$.

The Taylor series of the complex sine isTherefore, for $z\ne 0$,and the right side defines a holomorphic function at $z=0$ with value $1$. Thus the apparent singularity is a removable singularity.

The resulting power series isThe ratio test shows convergence for every $z\in \mathbb{C}$, so its radius of convergence is

For $t\to +\infty$ with $qE>0$, an asymptotic expansion givesso the longitudinal velocity tends to the speed of light. More generally, its limiting sign is the sign of $qE$.

For small $t$, the Taylor series of the square root givesIn the nonrelativistic limit ${\mathcal{E}}_0\simeq m{c}^2$, this becomesThe transverse motion is then uniform:Eliminating $t$ gives the nonrelativistic parabolic trajectory

For $f\notin E$ and rational $q\in (0,1)$, every $M$ admits $m\ge M$ withConsequentlyBy the Cauchy-Hadamard theorem, the Taylor series of $f$ at $q$ has radius of convergence zero.

Suppose some Taylor series represented $f$ throughout a neighborhood of a point $x\in [0,1]$. That neighborhood contains a rational $q\in (0,1)$. A function represented locally by a convergent power series is a real analytic function, and re-expanding that series about $q$ gives the Taylor series of $f$ at $q$ a positive radius of convergence. This contradicts the preceding conclusion. Thus $f$ agrees with no Taylor series on a neighborhood of any point; it is a nowhere-analytic generic smooth function.

The function $U(s)=V_0{e}^{-2s}$ is strictly convex. Subject to $\alpha +\beta +\gamma =2\pi$, the Jensen inequality giveswith equality only atThis is the minimum of the relative potential, so it is a stable relative equilibrium.

Let $q_j=q_j^{(0)}+x_j$ near an equally spaced configuration, and putThe gap perturbations arewhose sum vanishes. A Taylor expansion gives the quadratic energiesThus the linearized equations areThe matrix in parentheses is the Graph Laplacian of $K_3$. Its constant eigenvector has eigenvalue zero, while every vector whose components sum to zero has eigenvalue three. One orthonormal set of three normal modes and their angular frequencies is thereforeThe zero mode is rigid rotation. The positive and equal squared frequencies on the two-dimensional relative subspace prove stability and agree with the general normal modes of three equal masses with a symmetric gap potential.

A one-parameter exponential family with natural statistic $Y$ has density or probability mass functionwhere $\theta$ is the natural parameter of an exponential family and $K$ is the cumulant function of an exponential family. Its mean parameter is

The exponential-family deviance from ${\theta }_1$ to ${\theta }_2$ is twice the Kullback-Leibler divergence:The carrier $h(Y)$ cancels from the likelihood ratio, so, with ${\mu }_1={\mathbb{E}}_{{\theta }_1}Y$,

For the Poisson distribution with mean $\mu$,ThereforeWriting ${\mu }_2={\mu }_1+\delta$ and applying the Taylor series of $\log (1+\delta /{\mu }_1)$ givesso the second-order approximation is

A transformation of phase space is canonical when it preserves the symplectic form, or equivalently all Poisson brackets. For one degree of freedom this isequivalently $\{Q,P{\}}_{q,p}=1$.

Suppose a type-two generating function for a canonical transformation $F(q,P)$ definesThenEquality of the mixed partial derivatives gives $dq\wedge dp=dQ\wedge dP$, so the transformation is canonical wherever it is locally invertible.

Forthe fundamental theorem of calculus and differentiation under the integral sign giveThusand in particular $P=(p^2+q^2)/2$.

For the unit-frequency simple harmonic motion, the transformed Hamiltonian is simply $H=P$. Hamilton's equations becomeThereforeand transformation back gives the familiar solution

Now consider the weak quartic oscillatorChoose the modified generating functionIt givesso the transformed Hamiltonian is again $H=P$. Hence $P=E$ and $Q=t-t_0$. Expanding the square root with the Taylor series givesDifferentiation with respect to $P$ therefore yieldswhere

Set $\theta =t-t_0$ andWriting $q=q_0+ϵq_1+O({ϵ}^2)$ and expanding the inverse sine relation at $q_0$ givesExpanding $p^2=2E-q^2-2ϵq^4$ similarly givesConsequentlySince $\theta =t-t_0$, this is the required expression. The formula is understood away from turning points where $p_0=0$ and the ratio $q/p$ itself is singular.

The Taylor series of the shock density ratio at $\beta =0$ isFor a reversible adiabatic process, $p/p_0=(\rho /{\rho }_0{)}^{\gamma }$, so the density ratio predicted by the ideal-gas adiabat isEquations (1) and (2) agree through order ${\beta }^2$. Thus a weak shock satisfies the adiabatic pressure-density relation with an error of order $O({\beta }^3)$, which in particular establishes the requested accuracy through $O({\beta }^2)$.

The Taylor series at $x=0$ isSubstituting $x=Jqm/T$ givesThe quartic coefficient is positive, while the quadratic coefficient changes sign atThis is therefore a continuous mean-field phase transition.

Because $R=O(ϵ)$ andthe Taylor series gives $f(R)=R+O({ϵ}^3)$ and $f^{\prime }(R)=1+O({ϵ}^2)$. The extra derivative terms in the f(R) equation are therefore at least second order. At first order the theory reduces to the Linearized Einstein equationsIn harmonic coordinates, equivalently the Lorenz gauge in linearized gravity ${\partial }_{\mu }{\bar{h}}^{\mu }{}_{\nu }=0$, the trace-reversed metric perturbationsatisfiesThus Linearized f(R) gravity about Minkowski spacetime has the same weak-field predictions as general relativity under these assumptions. Differences can first appear through nonlinear curvature terms or in backgrounds about which $f^{\prime \prime }(R)$ is nonzero.

The required Taylor series giveSeek $\beta =aϵ+b{ϵ}^2+O({ϵ}^3)$. Substitution in the exact equation from part (iii) gives successivelyThus $a=-7/12$ and $b=7/12$, so

Choose a compact interval around $x_0$ and its constant $C$. For $|h|<1/C$, the Lagrange remainder after order $n$ is at most $(C|h|{)}^{n+1}\to 0$. Thus the Taylor series with $a_n=f^{(n)}(x_0)/n!$ equals $f(x_0+h)$.

Let $\bar{\rho }(t)$ be the homogeneous density. A Lagrangian coordinate volume $d^{3}q$ contains the same mass as its image $d^{3}x$ under the Zeldovich approximation, so mass conservation givesSince $1+\delta =\rho /\bar{\rho }$, the density contrast is

For $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor isEquality of mixed partial derivatives makes it a real symmetric matrix. The real spectral theorem therefore supplies an orthonormal eigenbasis and real eigenvalues, which we denote by $-{\lambda }_1,-{\lambda }_2,-{\lambda }_3$. In that basis,and henceBefore the first crossing all factors are positive, allowing the absolute values to be omitted.

When $1-D{\lambda }_i=0$, the map loses rank in the corresponding principal direction. Its Jacobian determinant vanishes, trajectories meet, and shell crossing creates a cosmological caustic. The single-stream pressureless density formally diverges and the approximation no longer describes the subsequent multistream dynamics. If only one ${\lambda }_i$ is positive, one axis first collapses while the other two remain extended, producing a sheet or pancake. Collapse along a second and then a third principal axis produces filaments and nodes. Spatial variation of the eigenvalues joins these objects into the cosmic web around underdense voids.

The approximation succeeds because it reproduces linear growing-mode evolution exactly, preserves the initial tidal displacement and anisotropic collapse, and follows matter along nearly inertial comoving trajectories instead of expanding only the density at a fixed point. Large scales remain weakly nonlinear and are insensitive to the detailed dynamics after crossing. Its limitations begin at shell crossing: it permits streams to pass through one another, cannot produce virialized halos, and omits velocity dispersion, vorticity, gas pressure, shocks, feedback, and strongly nonlinear self-gravity.

For the one-dimensional displacement,so mass conservation givesThe earliest crossing occurs where the cosine is maximal:for integers $n$. The denominator vanishes at these isolated points.

Write $u=q-q_n$ at $t=t_c$. The Taylor series givesThus $|u|\sim (6|x-x_n|/k^2{)}^{1/3}$ andThe first caustic is therefore a cubic cusp with

The lowest positive root belongs to the symmetric family and lies in $(0,\pi /2)$; all antisymmetric roots are at least $\pi$. Let $\alpha =m/(2\rho \ell )\gg 1$. The equation $\cot z=\alpha z$ forces its first root to tend to zero: for any fixed $\epsilon >0$, the right side eventually exceeds $\cot \epsilon$ at $z=\epsilon$, so the decreasing-minus-increasing difference has its zero below $\epsilon$.

The Taylor series $\cot z=z^{-1}-z/3+O(z^3)$ then givesSince $\omega =cz/\ell$ and $c=\sqrt{T/\rho }$, the lowest angular frequency isPhysically, at such a low frequency each half-string is nearly straight. If the mass moves up by $Y$, the two slopes are approximately $Y/\ell$ and $-Y/\ell$, so the restoring force is $-2TY/\ell$. The resulting harmonic oscillator equation, $m\ddot{Y}+(2T/\ell )Y=0$, has exactly the leading angular frequency above. The next correction from the same expansion is $\omega /{\omega }_0=1-\rho \ell /(3m)+O((\rho \ell /m{)}^2)$.

Integrate the geometric series for $1/(1+x)$ from zero to $x$. The first three nonzero terms areThe resulting Taylor series converges for $|x|<1$.

For fixed real $a\ne 0$, write the square root as $|a|\sqrt{1+x^2/a^2}$ and use the binomial series with exponent $-1/2$:The absolute values are necessary for negative $a$. This Taylor series is valid for $|x|<|a|$. If $a=0$, the quotient is $x/|x|$ for $x\ne 0$; it has no continuous extension at zero and therefore has no Taylor expansion about zero.

Use the real logarithm near zero and subtract its two Taylor series:Thus the first three nonzero terms are

For an $N+1$ times continuously differentiable function, Taylor's theorem giveswhere $\xi$ lies between $x_0$ and $x$. An infinite Taylor series represents the function when its remainder tends to zero; this holds locally for an analytic function, rather than for every smooth function automatically.

The Taylor expansion about $a$ has coefficients given by the successive derivatives at $a$:When $f$ is $N+1$ times continuously differentiable on the interval between $a$ and $x$, Taylor's theorem gives $R_N(x)=f^{(N+1)}(\xi )(x-a{)}^{N+1}/(N+1)!$ for an intermediate point $\xi$. If the remainders tend to zero, this yields the infinite Taylor series $f(x)={\sum }_{k=0}^{\infty }{f}^{(k)}(a)(x-a{)}^k/k!$. The three functions below are real analytic near zero after taking the continuous extension in part (i).