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searching for Euler's identity 8 found (61 total)

Homogeneous polynomial (1,039 words) [view diff] exact match in snippet view article find links to article

{d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.} Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if P is a homogeneous polynomial

Integration using Euler's formula (1,136 words) [view diff] exact match in snippet view article find links to article

to use a half-angle formula to simplify the integrand. We can use Euler's identity instead: ∫ cos 2 ⁡ x d x = ∫ ( e i x + e − i x 2 ) 2 d x = 1 4 ∫ (

Sophomore's dream (1,448 words) [view diff] exact match in snippet view article find links to article

different domain (corresponding to changing variables by substitution), as Euler's identity itself can also be computed via an analogous integration by parts.

Trigonometric functions (10,679 words) [view diff] exact match in snippet view article find links to article

way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms

Polar coordinate system (6,700 words) [view diff] case mismatch in snippet view article find links to article

Cambridge University Press. ISBN 0-521-59461-8. Smith, Julius O. (2003). "Euler's Identity". Mathematics of the Discrete Fourier Transform (DFT). W3K Publishing

Discriminant (6,705 words) [view diff] exact match in snippet view article find links to article

partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials). In the case of a homogeneous bivariate

Riemann zeta function (10,620 words) [view diff] exact match in snippet view article find links to article

of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental

Continued fraction (8,701 words) [view diff] exact match in snippet view article find links to article

convergence problem. Here are two continued fractions that can be built via Euler's identity. e x = x 0 0 ! + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ = 1 + x