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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 and and then integrated.
Simplifying this further gives us the solution x = −3. It is easily checked that none of the zeros of x ( x + 1)( x + 2) – namely x = 0 , x = −1 , and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.
The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0. These reduction formulas can be used for integrands having integer and/or fractional exponents.
For example, x has a single (real) super-root if n is odd, and up to two if n is even. [citation needed] Just as with the extension of tetration to infinite heights, the super-root can be extended to n = ∞, being well-defined if 1/e ≤ x ≤ e.
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of ...
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
We can also use the bits of the exponent in left to right order. In practice, we would usually want the result modulo some modulus m. In that case, we would reduce each multiplication result (mod m) before proceeding. For simplicity, the modulus calculation is omitted here.
For, if one applies Euclid's algorithm to the following polynomials [2] + + + and + +, the successive remainders of Euclid's algorithm are +, +,,. One sees that, despite the small degree and the small size of the coefficients of the input polynomials, one has to manipulate and simplify integer fractions of rather large size.
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