borel transform sentence in Hindi
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- The integral form for the generalized Borel transform follows as well.
- Nachbin's theorem gives necessary and sufficient conditions for the Borel transform to be well defined.
- The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type.
- In particular, Nachbin's theorem may be used to give the domain of convergence of the "'generalized Borel transform "', given below.
- Suppose that the Borel transform converges for all real numbers to a function growing sufficiently slowly that the following integral is well defined ( as an improper integral ), the "'Borel sum "'of " A " is given by
- More precisely, the series for the Borel transform converges in a neighborhood of the origin, and can be analytically continued to the positive real axis, and the integral defining the Borel sum converges to " f " ( " z " ) for " z " in the region above.
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