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The nitrate ion, ${\mathrm{NO}}_3^{-}$, is an anion with one nitrogen connected to three oxygen chemical atoms and net electric charge $-1$ in units of elementary charge. Its 24 valence electrons admit three equivalent octet resonance structures; see Lewis structure for their construction and chemical resonance for their interpretation. The N–O connections are covalent bonds within the ion. See 7.2 Covalent Bonding.

A manganese-containing anion with net electric charge $-e$ and manganese oxidation state $+7$; $e$ is the positive elementary charge.

Chloride is the anion $\mathrm{C}{\mathrm{l}}^{-}$, with net electric charge $-e$. In the glycine titration model, it supplies the negative background electric charge introduced with hydrochloric acid.

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/3-6-cellular-differentiation

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/1-5-homeostasis

Cited passages: §1.5, Figure 1.10 and thermoregulatory examples following it; §1.5, Negative Feedback, Figure 1.10: sensor, control center and effector; §1.5, opening paragraphs and Negative Feedback: normal ranges and compensatory responses.

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/17-9-the-endocrine-pancreas

Cited passages: 17.9, delta cells and somatostatin; 17.9, glucagon signaling through cAMP and glycogen breakdown; 17.9, pancreatic islet cell types and regulation of blood glucose.

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/12-2-nervous-tissue

Cited passages: Anatomy and Physiology 2e, §12.2, Parts of a Neuron: nucleus, organelles, processes, branching axon, initial segment and terminals; Anatomy and Physiology 2e, §12.2, Types of Neurons and Figure 12.9: multipolar, bipolar and human pseudo-unipolar cells; sensory projection past the cell body.

Online text: chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Analytical_Sciences_Digital_Library/Courseware/Analytical_Electrochemistry:_Potentiometry/06_Experiments/01_Potentiometric_Titration_of_an_Unknown_Weak_Monoprotic_Acid

Suppose that, as $t\downarrow 0$,and that the corresponding Laplace integral has suitable integrability and growth control away from zero. Watson lemma gives the endpoint expansionas $x\to +\infty$. Each finite truncation of the local amplitude expansion produces the corresponding finite truncation and remainder of the integral expansion.

A function is piecewise continuous on a finite interval if it is continuous between finitely many breakpoints and has finite one-sided limits at those points and at the endpoints. On $[0,\infty )$, require this on every finite interval. A Heaviside step function is an example; a function with an infinite blow-up at an interior point is not. This condition guarantees local integrability while allowing jumps in an input.

Harmonic analysis decomposes functions into elementary oscillations, generalizing Fourier series and the Fourier transform. On a locally compact Abelian group the elementary oscillations are continuous unitary characters, and Haar measure supplies translation-invariant integration.