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Steady two-dimensional flow to wells in uniform background flow
Published in Mark Bakker, Vincent Post, Analytical Groundwater Modeling, 2022
In mathematical terms, this means that the stream function is the harmonic conjugate of the discharge potential (Strack, 1989). The complex potential Ω is a function of the complex coordinate ζ = x + iy, where i is the imaginary unit (i2 = −1). The letter is used for the imaginary unit in Python, but i is used in the mathematical equations in this book. The negative derivative of the complex potential is the complex discharge W(ζ)=−dΩdζ=Qx−iQy
Holomorphic and Harmonic Functions
Published in Steven G. Krantz, Complex Variables, 2019
Example 11 Consider the function u(x, y) = x2 − y2 − x on the square U = {(x, y) : |x| < 1, |y| < 1}. Certainly U is simply connected. And one may verify directly that Δu ≡ 0 on U. To solve for v a harmonic conjugate of u, we use the Cauchy–Riemann equations: ∂v∂y=∂u∂x=2x−1,∂v∂x=−∂u∂y=2y.
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
Given a harmonic function u(x, y), can one find a function v(x, y) such that f (z) = u(x, y) + iv(x, y) is differentiable? In this case, v is called the harmonic conjugate of u. The harmonic conjugate function.
Inverse resonance problem with partial information on the interval
Published in Applicable Analysis, 2022
Lung-Hui Chen, Tzong-Mo Tsai, Chung-Tsun Shieh
( Nevanlinna–Levin) If the function is holomorphic and of exponential type in the half-plane and if (24) holds, then where is the harmonic conjugate of and are the zeros of the function in the half-plane ;where
Inverse phaseless scattering on the line with partial information
Published in Waves in Random and Complex Media, 2022
If the function is holomorphic and of exponential type in the half-plane and if (18) holds, then where , , is the harmonic conjugate of and are the zeros of the function in the half-plane ;where
On non-homogeneous Robin reflection for harmonic functions
Published in Applicable Analysis, 2022
Murdhy Aldawsari, Tatiana Savina
Let be a function, defined by formula (7), and let Γ be a segment of the x-axis, then the derivative of v with respect to x in the plane, is harmonic conjugate to the function defined by (5).