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The discussion revolves around mathematical series and their representations, particularly using binomial sums. The initial focus is on the series expansion of the function 1/sqrt(1 - x^2) and its implications for further calculations. The conversation includes deriving a complex expression involving the imaginary part of a function related to exponential terms. The participants are encouraged to proceed with the calculations based on the provided formulas. The exchange emphasizes the importance of mathematical rigor in understanding these series.
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chisigma
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anil86 said:
Wellcome on MHB anil86!... You can start from the binomial sum...
$\displaystyle \frac{1}{\sqrt{1 - x^{2}}} = 1 + \frac{1}{2}\ x^{2} + \frac{3}{8}\ x^{4} + ...\ (1)$
... and from (1) ...
$\displaystyle \frac{x}{\sqrt{1 - x^{2}}} = x + \frac{1}{2}\ x^{3} + \frac{3}{8}\ x^{5} + ... \ (2)$
... so that the sum of Your series is... $\displaystyle S = \text{Im} \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\}\ (3)$
Are You able to proceed?...
Kind regards
$\chi$ $\sigma$
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anil86
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chisigma
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chisigma said:
Is...
$\displaystyle 1 - e^{2\ z} = - e^{z}\ (e^{z} - e^{- z}) -> \sqrt{1 - e^{2\ z}} = i\ e^{\frac{z}{2}}\ \sqrt {e^{z} - e^{- z}}\ (1)$
... so that for $\displaystyle z = i\ \theta $ is...
$\displaystyle \text{Im}\ \{\frac{e^{i\ \theta}}{\sqrt{1 - e^{2\ i\ \theta}}}\} = \text{Im}\ \{ \frac{e^{i\ \frac{\theta}{2}}}{i\ \sqrt{e^{i\ \theta} + e^{- i\ \theta}}} \} = \frac{\sin \frac{\theta}{2} + \cos {\frac{\theta}{2}}}{2\ \sqrt{\sin \theta}}\ (2)$
Kind regards
$\chi$ $\sigma$
Obviously, there is something elementary I am missing here.
To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix
in the real plane; taking the transpose we get
which then corresponds to a-bi...
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