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Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.
For real non-zero values of x, the exponential integral Ei(x) is defined as = =. The Risch algorithm shows that Ei is not an elementary function.The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.
The exponential function is the sum of the power series [2] [3] ... Let b be a positive real number. The exponential function and ... List of integrals of exponential ...
Definition (3) presents a problem because there are non-equivalent paths along which one could integrate; but the equation of (3) should hold for any such path modulo . As for definition (5), the additive property together with the complex derivative f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} are sufficient to guarantee f ( x ) = e x ...
"An exponential method of numerical integration of ordinary differential equations". Communications of the ACM. 6 (8): 491– 493. doi: 10.1145/366707.367592. S2CID 18598461. Tokman, Mayya (October 2011). "A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK)". Journal of Computational Physics. 230 (24): 8762 ...
The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind [4] [5]: 113 ( ()) = and ( ()) = (). For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas .
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.