Gaussian curve, first derivative

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

Fig. 35. Characteristic line shape for a) Lorentzian and b) Gaussian absorption curve, together with functions for absorption and first derivative curves...

Fig. 4.5-1 The progress and titration curves, the Schwartz and Gran plots, and the first derivative of the titration curve, for the titration of 0.1 M acetic acid (p Ka = 4.7) with 0.1 M NaOH, without Gaussian noise added, na = 0. The first derivative is computed with a 13-point moving parabola.

ESR lines usually exhibit shapes very close to those of Gaussian or Lorentzian functions. The intensities of ESR lines may be obtained by integration of the full absorption curve, by two consecutive integrations of the first-derivative curve, or by the approximation... 

Fig. 6.7 Allowed intra-ligand transitions from x to V -type ligand orbitals for tris-chelate complexes with D3 symmetry. The circular dichroism has a lower right-circularly polarized (rep) band and an upper left-circularly polarized Ocp) band. This gives the CD spectmm the appearance of the first derivative of a Gaussian curve, with a negative part at longer wavelength and a positive part at shorter wavelength...

If we consider an absorption band showing a normal (Gaussian) distribution [Fig. 17.13(a)], we find [Figs. (b) and (d)] that the first- and third-derivative plots are disperse functions that are unlike the original curve, but they can be used to fix accurately the wavelength of maximum absorption, Amax (point M in the diagram). 

The methods in Section 3.3.1 are concerned primarily with removing noise. Most methods leave peakwidths either unchanged or increased, equivalent to blurring. In signal analysis an important separate need is to increase resolution. In Section 3.5.2 we will discuss the use of filters combined with Fourier transformation. In Chapter 6 we will discuss how to improve resolution when there is an extra dimension to the data (multivariate curve resolution). However, a simple and frequently used approach is to calculate derivatives. The principle is that inflection points in close peaks become turning points in the derivatives. The first and second derivatives of a pure Gaussian are presented in Figure 3.10. 

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