Topicos De Estadistica Descriptiva Y Probabilidades Maximo Mitac !!TOP!!



 
 
 
 
 
 
 

Topicos De Estadistica Descriptiva Y Probabilidades Maximo Mitac

pH-modulated synthesis of cyclic peptides containing thiazole moieties by fungal membrane protease I.
pH-modulated synthesis of cyclic peptides was achieved by the combined use of a fungal membrane protease I from Rhizomucor miehei and cyclic anhydrides. The cyclization proceeded by catalytic peptide bond formation, and the results indicated that an amphipathic domain was necessary for the cyclization. Among the cyclic peptide analogues obtained, the thiazole peptides were especially attractive in terms of properties such as natural peptide mimics, and the structural analyses were subsequently performed on several cyclic peptides. The results revealed that thiazole derivatives participated in hydrophobic interactions and pi-cation interactions, thereby playing a key role in the binding of biomolecules.Q:

Calculate $\int_{ -\infty}^{\infty} \frac{e^{ -x^2}(1+x)}{1+x^2} dx$

I was wondering how to calculate this integral using the complex functions. Please explain.
$$\int_{ -\infty}^{\infty} \frac{e^{ -x^2}(1+x)}{1+x^2} dx$$

A:

Define $f(z) = e^{ -z^2}(1+z)$ with $z=x+iy$.
Then the integral you wish to calculate is
$$\int_{ -\infty}^{\infty} \frac{e^{ -x^2}(1+x)}{1+x^2} dx = 2\pi \operatorname{Re} \int_{ -\infty}^{\infty} \frac{e^{ -z^2}(1+z)}{1+z^2} \, dz.$$
Now, by the Cauchy integral formula,
$$f'(z) = 2\pi i e^{ -z^2}(1+z).$$
Taking the real part of this gives
$$\operatorname{Re}f'(z) = 2\pi \operatorname{Re} \left( 1+z \right) e^{ -z^2} = \pi e^{ -z^2}

https://colab.research.google.com/drive/1rT3CmfmWeoWBzfVYce8BdWY7FHSd1V7s
https://colab.research.google.com/drive/1nhVY8RxiYo_6UUDgnuZiT0RUwFZ8DrDH
https://colab.research.google.com/drive/1uM5Wll-pC6zkVqIaZ8UOnubyva7f4x2W
https://colab.research.google.com/drive/1Rv8fY__b30zeQoUEgRxc3xdjjq_LNrRn
https://colab.research.google.com/drive/1DOQx4x7zMhds0qJgfPjM07ZYRRZsi1Cx

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