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  2. Argument (complex analysis) - Wikipedia

    en.wikipedia.org/wiki/Argument_(complex_analysis)

    Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in ...

  3. Imaginary unit - Wikipedia

    en.wikipedia.org/wiki/Imaginary_unit

    The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication .

  4. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative , commutative , and ...

  5. Complex conjugate - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate

    In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}

  6. Imaginary number - Wikipedia

    en.wikipedia.org/wiki/Imaginary_number

    An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]

  7. Runge–Kutta methods - Wikipedia

    en.wikipedia.org/wiki/Runge–Kutta_methods

    The numerical solution to the linear test equation decays to zero if | r(z) | < 1 with z = hλ. The set of such z is called the domain of absolute stability. In particular, the method is said to be absolute stable if all z with Re(z) < 0 are in the domain of absolute stability. The stability function of an explicit Runge–Kutta method is a ...

  8. List of Runge–Kutta methods - Wikipedia

    en.wikipedia.org/wiki/List_of_Runge–Kutta_methods

    Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; [6] the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously. The simplest method from this class is the order 2 implicit midpoint method.

  9. Complex plane - Wikipedia

    en.wikipedia.org/wiki/Complex_plane

    The natural way to label θ = arg(z) in this example is to set −π < θ ≤ π on the first sheet, with π < θ ≤ 3π on the second. The imaginary axes on the two sheets point in opposite directions so that the counterclockwise sense of positive rotation is preserved as a closed contour moves from one sheet to the other (remember, the ...

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