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  2. Contour integration - Wikipedia

    en.wikipedia.org/wiki/Contour_integration

    However, the important thing to note is that z 1/2 = e (Log z)/2, so z 1/2 has a branch cut. This affects our choice of the contour C. Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.

  3. Branch point - Wikipedia

    en.wikipedia.org/wiki/Branch_point

    In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut. A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience. The logarithm has a jump discontinuity of 2 π i when crossing the branch cut. The logarithm can be made ...

  4. Exponential integral - Wikipedia

    en.wikipedia.org/wiki/Exponential_integral

    The sum converges for all complex , and we take the usual value of the complex logarithm having a branch cut along the negative real axis. This formula can be used to compute E 1 ( x ) {\displaystyle E_{1}(x)} with floating point operations for real x {\displaystyle x} between 0 and 2.5.

  5. Principal branch - Wikipedia

    en.wikipedia.org/wiki/Principal_branch

    A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π. These are the chosen principal values. This is the principal branch of the log function. Often it is defined using a capital letter, Log z.

  6. Complex logarithm - Wikipedia

    en.wikipedia.org/wiki/Complex_logarithm

    For example, the principal branch has a branch cut along the negative real axis. If the function L ⁡ ( z ) {\displaystyle \operatorname {L} (z)} is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like Log ⁡ z {\displaystyle \operatorname {Log} z ...

  7. Riemann–Hilbert problem - Wikipedia

    en.wikipedia.org/wiki/Riemann–Hilbert_problem

    A crucial ingredient of the Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard Cauchy kernel (see Gakhov (2001) ; also cf. the scalar example below).

  8. Lambert W function - Wikipedia

    en.wikipedia.org/wiki/Lambert_W_function

    The branch point for the principal branch is at z = − ⁠ 1 / e ⁠, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W −1 and W 1. In all branches W k with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.

  9. Hypergeometric function - Wikipedia

    en.wikipedia.org/wiki/Hypergeometric_function

    For complex arguments z with | z | ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line z ≥ 1. As c → −m, where m is a non-negative integer, one has 2 F 1 (z) → ∞.