# Definition of complex plane in US English:

## complex plane

### noun

• An infinite two-dimensional space representing the set of complex numbers, especially one in which Cartesian coordinates represent the real and imaginary parts of the complex numbers.

• ‘He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane.’
• ‘There is also a whole field of mathematics called ‘complex analysis’ which studies functions and calculus on the complex plane rather than real numbers.’
• ‘As the Fundamental Theorem of Algebra clearly indicates, the complex plane rather than the real line is the proper place for the study of polynomials.’
• ‘The largest eigenvalue for each model is plotted in the complex plane.’
• ‘In the Standard Model the couplings between the light (up and down) quarks and the heavy (bottom and top) quarks are proportional to the lengths of the sides of the triangle in the complex plane.’
• ‘Cauchy developed his theory fitfully from the 1810s to the 1840s, and this version of his theorem is the last one, with the complex plane available as the site for C.’
• ‘In the complex plane, W has infinitely many branches.’
• ‘The Mandelbrot set is a connected set of points in the complex plane.’
• ‘Over much of the complex plane the function turns out to be wildly oscillatory, crossing from positive to negative values infinitely often.’
• ‘This limitation can be sidestepped by venturing off the real number line into the wilds of the complex plane.’
• ‘By considering the action of the modular group on the complex plane, Klein showed that the fundamental region is moved around to tessellate the plane.’
• ‘I learned about the complex plane at about the same time I learned of the quadratic formula.’
• ‘Montel also investigated the relation between the coefficients of a polynomial and the location of its zeros in the complex plane.’
• ‘More importantly, he discovered that all other solutions of the equation lie within a thin strip of the complex plane, between the real values and 1.’
• ‘Hence, the three eigenvalues would be complex numbers that lie somewhere in the complex plane within the areas defined by those circles.’