Kategoriarkiv: Imaginary numbers

Complex integral solved with Cauchy’s integral formula

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Publicerat i Calculus, Imaginary numbers, Uncategorized | Märkt , | Lämna en kommentar

Cauchy’s integralformula

Theorem Suppose U is an open subset of the complex plane C, f : U → C is a holomorphic function and the closed disk D = { z : | z − z0| ≤ r} is completely contained in U. Let … Läs mer

Publicerat i Calculus, Imaginary numbers | Märkt | Lämna en kommentar

Cauchy’s integral theorem

Cauchy‘s integral theorem states that an analytic function f(z) the line integral around a closed path C is zero. ∫f(z)dz = 0  . This means that the curve integrals over 2 curves with the same endpoints for an analytic function … Läs mer

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The Cauchy-Riemann equations

In order for a complex function of a  single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0. This condition leads to the CAuchy –Riemann equations- The … Läs mer

Publicerat i Advanced, Calculus, Imaginary numbers | Lämna en kommentar

de Moivre’s formula and complex-conjugation.

(e^ix )^n = cos(nx) + i sin(nx) is called de Moivre’s formula. The formula is named after the 17 th. century French huguenot mathematician Abraham de Moivre. Also the variable in of a function can be a complex number. f(z) … Läs mer

Publicerat i Gymnasiematematik(high school math), Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar

Alternative representations of complex numbers

As mentioned in the latest post any complex number may be represented by an arrow in the complex plane. This number is unambiguously described by two numbers: its real part x and its imaginary part y. z= x+iy. This is … Läs mer

Publicerat i Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar