The Microsoft devblogs are hosted by WordPress. Until recently, we haven’t been able to display equations using traditional math typography in our blogs except in images. Now we can embed LaTeX math which is a lot more accessible. One syntax for this is [ℒ]…[/ℒ], where the “…” has the desired LaTeX and the ℒ is the literal string “latex” (I can’t use the string “latex” itself since it would be interpreted as part of a LaTeX delimiter). Another syntax is “$ latex …$” except leave out the space I added following the $ to prevent it from turning into LaTeX. On web pages viewed in FireFox and Safari as well as in the latest versions of Chrome, Edge, or Opera, you can use native MathML. For many years, web pages could invoke MathJax to display MathML and LaTeX. I don’t think you can embed MathML in WordPress blog posts yet.

As an example of embedded LaTeX, several of my posts have included what I call the mode-locking integral

\(\displaystyle\frac{1}{2\pi}\int_0^{2\pi}\frac{d\theta}{a+b\sin\theta}=\frac{1}{\sqrt{a^2-b^2}}\)

The text for this equation is LaTeX, not an image! Very exciting!

You might wonder why I named it the “mode-locking integral”. It comes from being the solution to the mode-locking equation

\(\dot{\Psi}(t)=a+b \sin{\Psi(t)}\)

where \(\Psi\) is the relative phase angle between oscillating modes. The modes can be oppositely running waves in a ring laser, the oscillations of two pendulum clocks near one another on a wall, two nearby tuning forks, etc. For more detailed descriptions, see Section 7.5 Mode Locking in the book *Elements of Quantum Optics*, by Pierre Meystre and yours truly. The variable 𝑎 is the frequency difference between the oscillating modes in the absence of coupling, and the variable 𝑏 is the coupling coefficient. If |𝑎| > |𝑏|, the integral is valid. When |𝑎| ≤ |𝑏|, \(\dot{\Psi}=0\), which has the solution \(\Psi=-\sin^{-1}{\frac{a}{b}}\), and the modes are locked to the same frequency.

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