The natural logarithm has the parameter-integral representationFor $0<t<1$, the identity $(t^x-1)/\ln t={\int }_0^x{t}^{s}ds$ shows convergence: the absolute value is at most $|x|t^{\min (x,0)}$, which is integrable when $x>-1$. At $t=1$, the apparent singularity is removable, with limiting value $x$.

Write the left side as $I(x)$. On a compact parameter interval $c\le x\le d$ with $c>-1$, its differentiated integrand $t^x$ is bounded by the integrable function $t^c$. The dominated convergence theorem justifies differentiation under the integral sign, giving $I^{\prime }(x)={\int }_0^1{t}^{x}dt=1/(1+x)$. Since $I(0)=0$, the fundamental theorem of calculus gives $I(x)=\ln (1+x)$. At $x=-1$, the original improper integral diverges as $-{\int }_{0}dt/[t|\ln t|]$, and it also diverges for $x<-1$.

Integrating the geometric series for $1/(1+x)$ from zero to $x$ gives the displayed power series for $|x|<1$. It has radius of convergence one. At $x=1$ the alternating series test gives conditional convergence to $\ln 2$; at $x=-1$ the negative harmonic series diverges. Replacing $x$ by $-w$ gives $\ln (1-w)=-{\sum }_{n\ge 1}{w}^n/n$. An expression formed by substituting a nonlinear function for $w$ need not have the same convergence range after it is re-expanded as a Taylor series in the new variable.

Albert Abraham Michelson's 1907 lecture, Recent Advances in Spectroscopy, pp. 171–172, connects counting interference fringes with a reproducible length standard. Broad or multiple spectral lines obscured fringes at large path differences; red cadmium emission supplied a suitable source. His 1907 Nobel Prize in Physics recognized optical precision instruments and their spectroscopy and metrology applications.^[ref]

His authored account of the metre comparison at Sèvres describes a heated, evacuated cadmium discharge tube and a spectrometer selecting one spectral line. A movable reference mirror was compared with pairs of silvered parallel surfaces defining intermediate standards. Counted interference fringes established a short interval; repeated comparisons extended it to a decimetre and then a metre. White-light fringes located surface coincidence, while monochromatic fringes supplied counts and fractional orders. A thermally shielding box, firm pier, calibrated translation screw and tested thermometers controlled drafts, vibration, travel and temperature. Uniform temperature mattered in comparisons; the final standard required an absolute temperature reading. Optical microscopes made the final comparison to the marked metre bar, limiting the whole result.^[ref]

This is a measurement chain, rather than one enormous unchecked fringe count: short intervals establish whole orders; fractional-order readings refine each longer comparison. The optical coherence envelope helps locate white-light coincidence, whereas the carrier phase difference measures small departures from it. Repetition checks counting mistakes, but agreement alone cannot exclude shared systematic errors. The general lesson of metrology is to preserve the link from the optical ruler through each transfer and control to the final length.

A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index $n$ along the moving arm, and fixed vacuum wavelength ${\lambda }_0$. Write $\lambda ={\lambda }_0/n$ for the wavelength in that medium and take $\Delta L$ positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by $\Delta L$ adds $\Delta L$ before reflection and $\Delta L$ afterward. One bright-to-bright cycle means mirror travel $\lambda /2$; bright-to-dark means $\lambda /4$, half a cycle. All lengths must use compatible units, and $\Delta N$ is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.

For an original hypothetical example, suppose $\lambda =600\mathrm{nm}$ in the actual surrounding medium and a continuous scan gives $40.25$ cycles, including the final fractional cycle. Then $\Delta L=(600\mathrm{nm})(40.25)/2=12075\mathrm{nm}=12.075\mu \mathrm{m}$. With the wavelength treated as exact, a fringe-reading measurement uncertainty of $0.02$ cycle contributes $6\mathrm{nm}$ to the displacement measurement uncertainty. An uncertainty of $0.01\mathrm{rad}$ in a local phase difference instead corresponds to $600(0.01)/(4\pi )\mathrm{nm}\approx 0.477\mathrm{nm}$. This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.

To separate the oscillating readout from its optical coherence envelope, use the local modelHere $I$ is detected light intensity, $\bar{I}$ is the mean level, ${\varphi }_b$ is a fixed bias wave phase, and $\mathcal{V}$ is interferometric visibility. A fine scan of $D$ measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through $\mathcal{V}(D)$. The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known $\mathcal{V}$, one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.

The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving $\mathcal{V}(D)=\exp (-|D|/L_c)$ with $L_c=1\mathrm{mm}$. In vacuum, arm mismatches $|L_2-L_1|=0.60\mathrm{mm}$ and $0.10\mathrm{mm}$ give $\mathcal{V}=e^{-1.2}\approx 0.301$ and $e^{-0.2}\approx 0.819$. Matching the arms improves the phase difference readout without changing its ${\lambda }_0/2$ travel per cycle. A fine mirror scan of one cycle changes $D$ by only $600\mathrm{nm}$ for the stipulated $600\mathrm{nm}$ source, much less than $L_c$. These are calculated values, not historical data.

A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where $|dI/d\varphi |=\bar{I}\mathcal{V}$ is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.

Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed ${\lambda }_0$, the first-order change of the moving arm's round-trip path is $\Delta D\approx 2n\Delta L+2L\Delta n$. Thus an unrecognized $\Delta n$ mimics $\Delta L_{\mathrm{app}}=L\Delta n/n$. A hypothetical $L=0.10\mathrm{m}$, $n\approx 1$, and $\Delta n=10^{-7}$ give $10\mathrm{nm}$ apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.

If a diagonalizable matrix has distinct eigenvalues $r,s$, its spectral projections of a diagonalizable matrix are $(A-sI)/(r-s)$ and $(rI-A)/(r-s)$. Raising the diagonal form to a nonnegative integer power gives the displayed formula. Equivalently, interpolate the scalar function $t^n$ by a degree-one polynomial at $r,s$, then apply it to $A$. The diagonalizability assumption matters: knowledge of two eigenvalues alone does not exclude nontrivial Jordan blocks.

A diagonal matrix entry uses the same row and column index. The diagonal entries of a lower triangular matrix determine its determinant by their product.

A device with a reference magnetic layer whose magnetization is comparatively fixed and another layer whose magnetization responds more readily to an applied field. The resistance reads their relative alignment. Fert's account describes how such giant magnetoresistance sensors entered hard-disk read heads in 1997.^[ref]

Giant magnetoresistance connects the relative magnetizations of adjacent magnetic layers to their electrical transport. In a suitable magnetic-metal/nonmagnetic-metal stack, aligning the layers can leave one spin channel with an easy conducting path through both magnetic layers; opposite alignments make each channel encounter a more resistive layer. Albert Fert's account traces the spin-dependent-conduction idea to Nevill Mott and distinguishes the later device geometries.^[ref]

The beauty is that a magnetic arrangement becomes an electrical readout. Preparing reproducible layers, establishing their magnetizations, and measuring electrical resistance are separate parts of that inference. The Fe-Cr giant magnetoresistance experiments show how those parts were joined; the two-current resistor model isolates a simple transport mechanism under stated assumptions.

A dependence of electrical resistance on the angle between electric current and magnetization.

Consider isotropic electrons and electron holes with equal density $n$, charges $\mp e$, equal positive charge carrier mobility $\mu$, and single-species zero-field electrical conductivity ${\sigma }_0=ne\mu$. In a transverse magnetic field, opposite Hall conductivity components cancel, while diagonal components add to $2{\sigma }_0/[1+(\mu B{)}^2]$. Inverting gives ${\rho }_{xy}=0$ and ${\rho }_{xx}={\rho }_{yy}=[1+(\mu B{)}^2]/(2{\sigma }_0)$. Its large-field growth is ${\sigma }_0{B}^2/(2n^2{e}^2)$. The total zero-field resistivity is $1/(2{\sigma }_0)$, half the single-species resistivity; confusing these quantities introduces a factor-of-two normalization error.

If one temperature-dependent relaxation scale controls a fixed carrier population and fixed orbital geometry, the magnetoresistance ratio can collapse onto a function of $B/\rho (0,T)$. This is Kohler's rule. For the equal-mobility compensated two-carrier magnetoresistance model, $[\rho (B)-\rho (0)]/\rho (0)=(\mu B{)}^2$ and $\mu =[2ne\rho (0){]}^{-1}$. Therefore plotting against $[B/\rho (0){]}^2$ gives a common straight line of slope $1/(4n^2{e}^2)$. A collapse tests the scaling relation; it does not by itself prove this particular microscopic carrier model. Multiple independently varying scattering times or carrier populations can spoil it.

The Anderson positron track experiment identified a light positively charged particle, now called the Positron, by combining magnetic bending, loss of kinetic energy in lead, and particle range. Each observation answered a different question: bending constrained momentum, the lead supplied a direction of travel, and particle range tested the proposed mass.

Carl David Anderson's The Positive Electron (1933), pp. 491–493, describes the 1932 discovery in a vertical cloud chamber with a $15,000\mathrm{gauss}$ magnetic field. A $6\mathrm{mm}$ lead plate separated two photographed track segments. The effective viewing depth was $1\mathrm{cm}$ along the magnetic field, with $14\mathrm{cm}$ transverse diameter: this restricted out-of-plane motion. The photographic readout supplied curvature on both sides. Comparisons with electron tracks under the same conditions constrained ionization; specific ionization was not directly measured for these events. He tested accidental alignment of unrelated tracks and rejected a reversed electron interpretation requiring an increase of kinetic energy in lead. Assuming unit electric charge, the mass bound was about twenty electron masses, rather than an exact mass determination. The paper credits Robert Andrews Millikan with the design and Seth Henry Neddermeyer with assistance.^[ref]

Preparation used rapid piston expansion and a water–alcohol mixture in the cloud chamber; cooling permitted droplets to form along ionization trails. A camera photographed these droplets. Initially, expansions occurred at random, so many photographs contained no tracks. Later counter triggering improved the useful yield.^[ref]

In The Production and Properties of Positrons, his 1936 lecture, Carl David Anderson describes the chamber inside an electromagnet and explains why the lead plate was decisive: after losing kinetic energy, a particle bent more sharply. He describes a proton hypothesis predicting only $5\mathrm{mm}$ of particle range, against an observed path longer than $50\mathrm{mm}$. These are historical values from the lecture, not the stipulated example below. The point was the incompatibility of particle range with the momentum inferred from curvature, rather than positive electric charge alone.^[ref]

Consider lab-neutral parallel wires with stationary positive line density $+\lambda$ and electron line density $-\lambda$. Let wire 1's electrons move at $v_1$ along the wire and wire 2's at $v_2$. The conventional currents are $I_j=-\lambda v_j$. In the rest frame of wire 2's electrons, wire 1's transformed total linear charge density is ${\lambda }_1^{\prime }={\gamma }_2[0-v_2{I}_1/c^2]={\gamma }_2\lambda v_1{v}_2/c^2$. Its radial electric field is ${\lambda }_1^{\prime }/(2\pi {ϵ}_{0}d)$.

A wire-2 electron at rest there feels only an electric force. Transforming its transverse force back to the lab divides it by ${\gamma }_2$, and multiplying by the lab electron count per length $\lambda /e$ giveswith positive radial direction away from wire 1. The lab-stationary positive charges feel zero net electric force because wire 1 is neutral in their frame. Equal current directions attract; opposite directions repel. Transforming force on each carrier and then counting carriers in the specified frame avoids an incorrect extra length-contraction factor in force per unit length.

In the plane wave basis of a ring, $V(x)=-\lambda \cos (2\pi x/L)$ has matrix elements $V_{mn}=-(\lambda /2)({\delta }_{m,n+1}+{\delta }_{m,n-1})$. Near $\alpha =1/2$, degenerate perturbation theory on $n=0,1$ givesup to corrections from remote states. Thus the avoided crossing gap is $|\lambda |$ to first order when $|\lambda |\ll E_L$.

For $|\lambda |\gg E_L$, a parabolic approximation about a potential minimum instead gives a local quantum harmonic oscillator spacing $\sqrt{2|\lambda |E_L}$. A local gauge transformation removes the flux from this approximation. Actual ring levels retain exponentially small flux dependence through quantum tunnelling, while nonquadratic corrections to their mean energies are algebraic in $\sqrt{E_L/|\lambda |}$.

For reversible quasistatic magnetic circuits with flux linkages ${\lambda }_i$, currents $I_i$ and mechanical coordinate $q$, write $d{W}_B={\sum }_i{I}_{i}d{\lambda }_i-F_{q}dq$. The Legendre transform $W_B^{\prime }={\sum }_i{I}_i{\lambda }_i-W_B$ obeys $d{W}_B^{\prime }={\sum }_i{\lambda }_{i}d{I}_i+F_{q}dq$. Thus $F_q=(\partial W_B^{\prime }/\partial q{)}_{I_i}$, while $F_q=-(\partial W_B/\partial q{)}_{{\lambda }_i}$. In a linear reciprocal system $W_B^{\prime }=W_B=\frac{1}{2}{\mathbf{I}}^{\mathsf{T}}L(q)\mathbf{I}$ numerically, but the variables held fixed in their force derivatives differ. The coenergy formulation includes the effect of electrical sources maintaining the currents; it avoids reversing the sign of the magnetic force by differentiating stored energy under the wrong constraint.

Spatial variations in energy, including crystal defects, can trap a magnetic domain wall. This is magnetic domain-wall pinning. Increasing the applied magnetic field can release the wall into a collective jump.

Prepare two uniform specimens with the same material, direction and $A_0$, but different deforming lengths $L_1,L_2$. Keep grip engagement, adapters, loading window and measurement configuration equivalent. Determine $q_i=d{x}_i/dF$ separately by linear regression with its own displacement intercept. Subtracting the two series equations eliminates the common machine compliance:This is an inverse problem: one length supplies only one equation for two unknowns, while two distinct lengths separate the length-dependent specimen contribution from the constant contribution. In a graph of compliance against length, the slope is $1/(E{A}_0)$ and the vertical intercept is $C_m$. It is the compliance, not $k_{\mathrm{obs}}$, that is linear in $L$.

The dimensional analysis is revealing: $[q_2-q_1]=m/N$, so $[(L_2-L_1)/(A_0(q_2-q_1))]=N/{\mathrm{m}}^2$. For $L_2>L_1$, the ideal model requires $q_2>q_1$. A negative fitted $C_m$ challenges the common-intercept assumptions or the measurements. Two lengths determine two parameters exactly, so they cannot test the assumed straight line; three or more lengths allow regression residuals to reveal departures. Nearly equal lengths amplify slope errors through the small difference $q_2-q_1$. If the actual apparatus compliances differ by $\Delta C_m$, the inferred length slope becomes $1/(E{A}_0)+\Delta C_m/(L_2-L_1)$.

A logical implication holds whenever its premise is false. For partial correctness, an execution that never terminates supplies no final state violating the postcondition.

A log-log plot uses logarithmic scales on both axes. For a positive power law $y=C{x}^p$ with $x,C>0$, positive references $x_0,y_0$ giveIts transformed slope is the exponent $p$ when both axes use the same logarithm base and the slope is measured in logarithmic coordinate units. For example, $y=2x^2$ takes the values $2,8,32$ at $x=1,2,4$. Every doubling of $x$ quadruples $y$, giving slope two on a log-log plot.

This differs from a semi-log plot: a power law has constant proportional response to a proportional change of input, whereas exponential growth has constant proportional response to an additive change of input. Both can appear straight with appropriate axes, so one must read the scale labels before interpreting a slope.

A semi-log plot uses a logarithmic scale on one axis and a linear scale on the other. Which axis is logarithmic matters. Plotting $x$ against $\log y$ makes positive exponential functions appear as straight lines; plotting $\log x$ against $y$ instead makes an affine function of a logarithm appear straight. The plotting operations change positions while retaining the original tick values.^[ref]

Take the original dimensionless mathematical model $y(t)=3e^{0.4t}$ for $0\le t\le 8$. It has exponential growth, with $y(0)=3$, $y(4)\approx 14.8591$, and $y(8)\approx 73.5976$. Taking a base-ten logarithm givesThe slope in these transformed positions is $0.4/\ln 10\approx 0.173718$, although the original relative growth rate $y^{\prime }/y$ is $0.4$. Equal steps of two in $t$ multiply $y$ by $e^{0.8}\approx 2.22554$. The corresponding ordinary gains increase, so the left plot bends while the right plot is straight.

One practical use is checking a proposed exponential distribution for failure times. For rate $\lambda >0$ and $t\ge 0$, its cumulative distribution function is $F(t)=1-e^{-\lambda t}$, so the reciprocal survival function $y=1/[1-F(t)]$ satisfies ${\log }_{10}y=\lambda t/\ln 10$. Estimated probabilities can therefore be placed on a logarithmic vertical axis and compared with a straight line. Agreement supports compatibility with the chosen statistical model; a roughly straight finite sample does not establish a unique physical mechanism.^[ref]