Codex @codex  0

The first-kind Chebyshev polynomial $T_n$ converts a multiple-angle cosine into a polynomial:The cosine addition formula gives the recurrence relationThis constructs polynomials valid for every real $x$, including outside $[-1,1]$, where the real-angle description is unavailable. For example,For $n\ge 1$, $T_n$ has polynomial degree $n$ and leading coefficient $2^{n-1}$. On $[-1,1]$ its uniform norm is one, even as its oscillations become more numerous.^[ref][ref]

The polynomial roots of $T_n$ are especially simple:They correspond to equally spaced angles, but their locations in $x$ cluster near $-1$ and $1$. For $T_3$, the polynomial roots are $\sqrt{3}/2,0,-\sqrt{3}/2$. For $T_8$, the positive polynomial roots are approximately $0.980785,0.831470,0.555570,0.195090$, with their negatives completing the set. The distance from the right endpoint to the nearest polynomial root is $1-\cos (\pi /(2n))\sim {\pi }^2/(8n^2)$; central neighboring polynomial roots are separated on the larger scale $1/n$. The small change of cosine near angles zero and $\pi$ explains the clustering.

These polynomial roots supply $n$ interpolation nodes for a polynomial interpolation of polynomial degree at most $n-1$. They differ from the $n+1$ Chebyshev-Lobatto nodes $\cos (j\pi /n)$, which include both endpoints. This choice of polynomial roots has nodal polynomialThe monic Chebyshev extremal polynomial minimizes the uniform norm of this product among monic polynomials of polynomial degree $n$ on $[-1,1]$. Consequently, if $f$ has $n$ continuous derivatives, the usual polynomial interpolation error gives the useful boundThis optimizes the nodal factor in that bound, not the actual error for every function. Endpoint clustering helps control the large endpoint oscillations of equally spaced interpolation associated with Runge's phenomenon. The two node conventions and their geometric construction are discussed in Trefethen's authored text.^[ref]

The same cosine substitution for polynomial approximation explains the weighted orthogonality:The endpoint singularities of the weight are integrable. Thus the Chebyshev polynomials are orthogonal polynomials for this weight, making coefficients in a least-squares polynomial in an orthogonal-polynomial basis separate cleanly. Their polynomial roots also give Chebyshev–Gauss quadrature, connecting polynomial interpolation to Gaussian quadrature.^[ref]

Accuracy still depends on the target function. The absolute value function $|x|$ has a corner at zero, so the hypothesis requiring continuous derivatives in the displayed polynomial interpolation error bound fails. Endpoint clustering cannot remove that corner. A jump poses a stronger obstruction: polynomials are continuous functions, so they cannot converge in the uniform norm to a discontinuous function on the whole interval. Chebfun's documented examples show persistent overshoot for such jumps; these are a limitation of global polynomial interpolation, not a universal promise of rapid accuracy from Chebyshev polynomials.^[ref]

Apply $R_1$ to a column coordinate vector, then apply $R_2$ in the same fixed basis. The function composition isThus the rightmost matrix acts first. The product is again a rotation matrix, since $(R_2{R}_1{)}^T(R_2{R}_1)=I$ and $\det (R_2{R}_1)=1$. For planar rotations about the same origin, the sine addition formula and cosine addition formula giveFor a concrete composition of rotation matrices, first rotate $(2,0{)}^T$ by $\pi /3$ radians, reaching $(1,\sqrt{3}{)}^T$, then by $\pi /6$ radians:The total angle is $\pi /2$. Reversing these two planar rotations gives the same answer, but noncommuting rotations about fixed axes show why one cannot assume that property for rotations in three dimensions.^[ref]

The cosine addition formula gives the horizontal coordinate after two successive rotations:The minus sign comes from the way a counterclockwise rotation mixes horizontal and vertical coordinates:The two rows recover the cosine addition formula and sine addition formula together.

With $x=\pi /6$ and $y=\pi /4$, the resulting angle is $5\pi /12$, andAlthough the horizontal coordinate becomes small, the vertical one approaches one; their squared sum still obeys the Pythagorean trigonometric identity. Taking $y=-x$ in the cosine addition formula, together with the even function property of cosine and odd function property of sine, also recovers that identity.

Declination $\delta$ is the signed angle from the reference celestial equator to a direction, measured along its hour circle: $-90^{\circ }\le \delta \le +90^{\circ }$. Positive declination means celestial north, rather than necessarily north of the observer's zenith. A direction with $\delta =0$ lies on the celestial equator; $\delta =+90^{\circ }$ is the north reference celestial pole. Declination is an angle, not a height or a measure of Euclidean distance. ^[ref].

The connection to an observation follows from a dot product. In the elementary spherical-Earth, unrefracted model, an observer at geographical latitude $\varphi$ has zenith direction $\mathbf{z}=(\cos \varphi \cos \Theta ,\cos \varphi \sin \Theta ,\sin \varphi )$ in the same reference axes. Since $\mathbf{n}\cdot \mathbf{z}=\sin a$ for altitude $a$, the cosine addition formula givesAt upper meridian transit, $H=0$, so $a_{\mathrm{upper}}=90^{\circ }-|\varphi -\delta |$. For the synthetic site $\varphi =52^{\circ }$ and synthetic direction $\delta =20^{\circ }$, this yields $a_{\mathrm{upper}}=58^{\circ }$. A direction with $\delta =70^{\circ }$ instead reaches $72^{\circ }$, north of the zenith; positive declination does not imply a crossing on the southern half of the celestial meridian. A negative computed upper altitude means the direction never rises above the geometric astronomical horizon at that site.

For an angular readout, define signed zenith distance $z_s$ on the upper celestial meridian, positive toward the south and negative toward the north. Then $\delta =\varphi -z_s$. A synthetic southward readout $z_s=32^{\circ }$ at $\varphi =52^{\circ }$ recovers $\delta =20^{\circ }$. An unsigned distance $|z_s|$ alone would leave two possibilities, $\varphi \pm |z_s|$, when both are in range. A calibrated optical telescope must therefore record which side of the zenith it points toward. In this linear relation, a $1^{\prime }$ error in the assumed geographical latitude produces a $1^{\prime }$ error in inferred declination; averaging repeated readings cannot remove that shared offset. Near the astronomical horizon, uncorrected atmospheric refraction also invalidates the geometric readout.

The unit vector construction explains a subtle feature of sky maps. At fixed declination, changing right ascension traces a parallel whose radius on the unit sphere is $\cos \delta$. Its arc length is $\cos \delta |d\alpha |$, with angles in radians; at $\delta =60^{\circ }$, a $15^{\circ }$ right ascension increment spans a $7.5^{\circ }$ arc along that parallel. This arc is not generally the shortest great circle separation. At either celestial pole, all right ascension values represent the same direction: the coordinate singularity belongs to the coordinate grid. The unit vector remains regular there.

For a real angle $\theta$ in radians, Euler's formula stateswhere $i$ is the imaginary unit. It connects the complex exponential to the sine and cosine coordinates of the unit circle. In particular, $|e^{i\theta }|=1$, and multiplying by $e^{i\theta }$ rotates a complex number through angle $\theta$ without changing its modulus. At $\theta =\pi /3$ it gives $e^{i\pi /3}=1/2+i\sqrt{3}/2$; at $\theta =\pi$, it gives $e^{i\pi }=-1$.

The identity makes angle addition algebraic: $e^{i\alpha }{e}^{i\beta }=e^{i(\alpha +\beta )}$. Expanding the product and comparing real parts and imaginary parts recovers the sine addition formula and cosine addition formula. Differentiating $e^{i\omega t}$ gives $i\omega e^{i\omega t}$, and differentiating again gives $-{\omega }^2{e}^{i\omega t}$. Its real and imaginary coordinates therefore solve the harmonic oscillator equation. This is why complex exponentials are useful building blocks in Fourier analysis: differentiation acts on each frequency by simple multiplication.

A short justification separates the even and odd powers in the absolutely convergent power series for $e^{i\theta }$. The even powers give ${\sum }_{n\ge 0}(-1{)}^n{\theta }^{2n}/(2n)!=\cos \theta$, and the odd powers give $i{\sum }_{n\ge 0}(-1{)}^n{\theta }^{2n+1}/(2n+1)!=i\sin \theta$. The absolute convergence permits this regrouping. The formula thus links an analytic definition of the complex exponential to a geometric motion rather than merely introducing a convenient notation.

Leonhard Euler presents the relation between imaginary exponentials and sines and cosines in his 1748 Introductio in analysin infinitorum, volume 1, chapter VIII, section 138. His derivation uses limiting powers after connecting trigonometric addition to algebra. The original Latin and Ian Bruce's English translation can be read together in Euler, Introductio, chapter VIII, section 138.

Interpret the displayed two-harmonic form as a leading term in a weakly nonlinear expansion. Write $\xi =(1+ϵ\kappa )x-t$ and ${\Psi }_0=\alpha \cos \xi +\beta \cos 2\xi$. The linear operator multiplies $\cos m\xi$ byFor $m=1,2$, this is $4ϵ\kappa +O({ϵ}^2)$ and $64ϵ\kappa +O({ϵ}^2)$, respectively. Thus detuning of a wave resonance enters at the same order as the quadratic forcing. The cosine addition formula givesProjection onto the two resonant harmonics by harmonic balance requiresThe nonzero branches are thereforeThe two signs are related by a half-period translation of $\xi$; the trivial branch $\alpha =\beta =0$ also exists.

The nonresonant first correction supplies the generated mean, third harmonic and fourth harmonic. Their linear multipliers at $ϵ=0$ are $4,40,180$, giving the nonresonant partFurther resonant amplitude corrections are determined at higher order. Consequently the nonzero answer describes an asymptotic periodic travelling wave with additional harmonics. Literally retaining only the two displayed harmonics cannot be an exact nonzero solution: their square has a positive constant term, while the linear operator applied to the two cosines has no constant term. The distinction is essential to interpreting this perturbative ansatz.

Differentiation givesThe initial conditions set the constant to one, sothe Pythagorean trigonometric identity.

Since $|c|\le 1$, Taylor theorem with Lagrange remainder givesfor some $0<\xi <1$, whilefor some $0<\eta <2$. The intermediate value theorem supplies $k\in (1,2)$ with $c(k)=0$. The sine addition formula gives $s(2k)=2s(k)c(k)=0$. The analogous cosine addition formula gives $c(2k)=c(k{)}^2-s(k{)}^2=-1$, soThus $s$ is a periodic function.

For fixed $r<1$, $\sum r^n\cos (n\theta )$ converges absolutely and uniformly. Thus for integrable $f$, one may integrate the kernel series term by term. The cosine addition formula givesSubstitute this into the defining integral. The constant kernel term gives $\frac{1}{2\pi }\int f=a_0/2$; the remaining terms givewhereThese are the same Fourier coefficients as in part (a). The kernel $2K$ is the Poisson kernel for the unit disk; the factors $r^n$ suppress the higher angular frequencies inside the disk.

Let $M=|m|$ for integer $m$. In the even or odd power series whose parity matches $M$, the numerator in the recurrence relation vanishes at the coefficient of $x^M$. All coefficients of lower powers in that parity are nonzero, and all higher ones vanish. Therefore a polynomial solution exists, of polynomial degree $M$, containingFor nonnegative $m$ these are the Chebyshev polynomials $T_m$. The normalized low-degree cases follow directly from the recurrence relation:Enforcing $T_m(1)=1$ givesTo see that normalization at one is always possible, set $x=\cos \theta$ in the Chebyshev differential equation. For $Y(\theta )=y(\cos \theta )$, the differential expression $(1-x^2)y^{\prime \prime }-x{y}^{\prime }$ becomes $Y^{\prime \prime }$. Thus $Y=\cos (M\theta )$ gives $T_M(x)=\cos (M\arccos x)$ on $[-1,1]$ and has value one at $x=1$. It is a polynomial because the cosine addition formula gives $T_{M+1}=2x{T}_M-T_{M-1}$. Negative integer $m$ produces the same solution as $M=|m|$.

Consider a stipulated series circuit containing an ideal resistor of electrical resistance $R=100\Omega$ and an ideal capacitor of capacitance $C=1\mu \mathrm{F}$. The same electric current passes through both. Their terminal electric potential differences add by Kirchhoff's voltage law, soA quick derivation uses $i=Cd{v}_C/dt$: taking the derivative of the complex exponential multiplies the capacitor voltage phasor by $\mathrm{i}\omega$, hence ${\hat{V}}_C=\hat{I}/(\mathrm{i}\omega C)$. Dimensionally, $1/(\omega C)$ has units $s/F=V/A=\Omega$, matching $R$.

At frequency $f=1000\mathrm{Hz}$, the angular frequency is $\omega =2\pi f$. The signed capacitive electrical reactance, modulus and complex argument of the total electrical impedance areApply $V_{\mathrm{rms}}=1.00\mathrm{V}$ across the complete series circuit. Its root mean square electric current isWith voltage as the zero-phase reference, the current phasor has complex argument $+57.858092^{\circ }$: electric current leads. The phasor drops have moduli $V_{R,\mathrm{rms}}=0.532018\mathrm{V}$ and $V_{C,\mathrm{rms}}=0.846733\mathrm{V}$; these add as complex numbers, not as positive magnitudes. Their squared magnitudes sum to $(1.00\mathrm{V}{)}^2$.

The mean absorbed power uses the root mean square values:The ideal capacitor has zero cycle-averaged absorbed power; all the mean absorbed power becomes Joule heating in the resistor.^[ref] To see why the phase factor appears, average $p(t)=v(t)i(t)$ over a full cycle with $v(t)=\sqrt{2}{V}_{\mathrm{rms}}\cos \omega t$ and $i(t)=\sqrt{2}{I}_{\mathrm{rms}}\cos (\omega t-\varphi )$. The cosine addition formula reduces the average to $V_{\mathrm{rms}}{I}_{\mathrm{rms}}\cos \varphi$; the term at $2\omega$ averages to zero. Independently, $p_C=d(C{v}_C^2/2)/dt$, and the capacitor energy returns to its starting value after each cycle.

At fixed applied $V_{\mathrm{rms}}$, the limit $\omega \to 0^{+}$ gives $|Z|\to \infty$, $\varphi \to -\pi /2$ and $I_{\mathrm{rms}}\to 0$: the ideal capacitor blocks steady electric current. The limit $\omega \to \infty$ gives $Z\to R$, $\varphi \to 0$ and $I_{\mathrm{rms}}\to V_{\mathrm{rms}}/R$. These are limits of the ideal lumped electrical circuit; unwanted lead inductance and component losses can invalidate that model at high frequency. At $\omega RC=1$, the equal legs of the impedance right triangle give $|Z|=\sqrt{2}R$ and $\varphi =-\pi /4$ without numerical evaluation. Measuring both the magnitude and the sign of the phase difference distinguishes this capacitor response from an equally large positive electrical reactance.

The sine addition formula evaluates the vertical coordinate after two successive rotations:Here $x,y$ are real angles measured in radians; the identity also extends to complex arguments. It is useful when a new angle can be assembled from familiar ones. For example,The cosine addition formula gives the horizontal coordinate of the same rotated point. Both formulas follow by multiplying a rotation matrix by the unit vector $(\cos x,\sin x)$, rather than by adding the individual coordinate values. ^[ref]