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Hyperbolic functions combine the exponential function to describe a hyperbola much as trigonometric functions describe a circle. Their two basic members are the hyperbolic cosine and hyperbolic sine, defined for every real number $t$ byTheir derivatives exchange the two functions:These formulas follow by applying differentiation to the exponential function definitions. ^[ref]

Adding and subtracting gives $\cosh t+\sinh t=e^t$ and $\cosh t-\sinh t=e^{-t}$. Multiplying these equalities proves the hyperbolic Pythagorean identity:Consequently the parametric equation $(x,y)=(\cosh t,\sinh t)$ traverses the entire right branch of the unit hyperbola $x^2-y^2=1$, where $x\ge 1$. For example, $t=0$ gives $(1,0)$, while $t=\pm 1$ gives approximately $(1.543,\pm 1.175)$. Compare $(\cos \theta ,\sin \theta )$ on the unit circle, where the squared coordinates add to one. The minus sign changes the geometry from a closed circle to an unbounded hyperbola.

The other members are ratios and reciprocals: hyperbolic tangent is $\tanh t=\sinh t/\cosh t$, hyperbolic cotangent is $\coth t=\cosh t/\sinh t$, and hyperbolic secant is $\mathrm{sech}t=1/\cosh t$; denominators must be nonzero. The hyperbolic sine and hyperbolic cosine both solve the second-order linear differential equation $f^{\prime \prime }=f$. In fact $f(t)=A\cosh t+B\sinh t$ has $f(0)=A$ and $f^{\prime }(0)=B$, making this basis convenient for an initial value problem. Their geometry also makes hyperbolic substitution useful for integrals containing $\sqrt{1+u^2}$, and the hyperbolic cosine gives the physical catenary of a hanging chain.

The hyperbolic cosine is entire and $\cosh 0=1$. Continuity makes it nonzero in a neighborhood of zero, so its reciprocal is holomorphic there and has a convergent Taylor series. Since $\cosh$ and its reciprocal are even, all odd coefficients vanish.

Multiply the proposed expansion by $\cosh z$:Coefficient comparison gives $c_2+1/2=0$ and $c_4+c_2/2+1/24=0$, henceThe zeros of $\cosh z$ solve $e^{2z}=-1$, so they are $z=i(\pi /2+k\pi )$, $k\in \mathbb{Z}$. The nearest ones are $\pm i\pi /2$, and they are simple because $\sinh$ is nonzero there. The hyperbolic secant has genuine poles at these points and is holomorphic in the disk of radius $\pi /2$. The radius of convergence of its Taylor series is therefore

For a real steady amplitude, integrate the conserved-field equation twice. Periodicity forces the integration slope to vanish, and the zero mean fixes the remaining constant. HenceSubstituting this into the steady amplitude equation gives the nonlocal ordinary differential equationFor $A=R\mathrm{sech}(cX)$, use the hyperbolic secant identity $A^{\prime \prime }=c^{2}A-2c^2{A}^3/R^2$. Vanishing of the two independent coefficients givesChoose $c>0$ and $R>0$; an overall negative sign is another solution. Thus $\kappa >1$. The exact average of the proposed profile over the finite interval isFor $cL\gg 1$, the hyperbolic tangent differs from one by $O(e^{-cL})$, so $cL⟨A^2⟩\simeq 2R^2$. Eliminating $R^2=2c^2/(\kappa -1)$ gives the requested leading relationEquivalently $c+c^{-1}=K$, with $K=4\kappa /[L(\kappa -1)]$. Positive real roots exist precisely when $K\ge 2$; they are $(K\pm \sqrt{K^2-4})/2$ and are reciprocal. Each candidate must additionally satisfy $cL\gg 1$. For $L>2$, the leading necessary range is $1<\kappa \le L/(L-2)$, with equality at the coalescence $c=1$. Away from this leading-order fold, the inequality is strict.

The strict finite-window condition can also be seen by retaining the average exactly. Its consistency relation is $c+c^{-1}=K\tanh (cL/2)$. Since $\tanh (cL/2)<1$, it implies $K>2$, and henceThe strict endpoint distinction is exponentially small in the large-$cL$ approximation; the leading polynomial alone permits its double root. When $0<L\le 2$, this algebra imposes no finite upper bound on $\kappa >1$. The printed interval therefore needs $L>2$: for example $L=1$, $\kappa =1.1$ gives a large positive root $c\simeq 43.98$ of the leading equation and $cL\gg 1$, despite the printed upper endpoint being negative.

Finally, the single localized pulse of a conserved-field convection model is an approximate periodic solution. Its values agree at the interval ends, but its derivatives have opposite signs there; their magnitude is exponentially small, $O(cR{e}^{-cL/2})$. Periodic-tail corrections are needed for an exact smooth periodic boundary condition. The range derived from the truncated profile is a consistency condition, not a proof that every permitted parameter gives an exact stable periodic pulse.

For fixed $x\ne 0$, the chain rule and the derivative of the hyperbolic secant giveThus the stated antiderivative holds, with its integration constant independent of $y$ but allowed to depend on $x$. At $x=0$ the integrand is zero and the antiderivative is simply constant; the displayed formula with $1/x$ is meant for nonzero $x$.

A probability density function must be nonnegative. The printed condition $s\ne 0$ is therefore too weak: for $s<0$ the given prefactor makes the function negative everywhere. The location-scale form of the logistic distribution requires $s>0$, which is assumed in the remaining parts.

Set $t=(x-\mu )/(2s)$, so $dx=2sdt$. The normalization of a probability distribution follows from the antiderivative of the squared hyperbolic secant:For completeness, the same expression with $s<0$ integrates to $-1$, since the transformed limits reverse. If a signed scale were desired, replacing its prefactor by $1/(4|s|)$ would give a valid density, but that is a correction to the printed formula.