Complex numbers, analytic functions, Cauchy-Riemann equations, contour integrals, residues and conformal maps.
39 cards · basic cards · AI-written, checked twice. Edit anything.
- What is the modulus of a complex number z = a + bi?
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|z| = sqrt(a^2 + b^2)- In polar form z = r*e^(i*theta), what do r and theta represent?
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r is the modulus, theta is the argument (angle from positive real axis)- State De Moivre's theorem
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(cos(theta) + i*sin(theta))^n = cos(n*theta) + i*sin(n*theta)- What is an analytic function?
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A function that is complex-differentiable in an open set- State the Cauchy-Riemann equations for f(x+iy) = u(x,y) + i*v(x,y)
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du/dx = dv/dy and du/dy = -dv/dx- What is a harmonic function?
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A twice continuously differentiable real function satisfying Laplace's equation: d^2u/dx^2 + d^2u/dy^2 = 0- State Cauchy's theorem
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If f is analytic on and inside a closed contour C, then the contour integral of f dz around C equals zero- State Cauchy's integral formula
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f(z_0) = (1/(2*pi*i)) * integral of f(z)/(z-z_0) dz around a closed contour containing z_0- Define a removable singularity
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A point z_0 where f is not defined but lim(z->z_0) f(z) exists and is finite- What is a pole of order n?
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A singularity z_0 where f(z) has the form A/(z-z_0)^n + (lower order terms), with A ≠ 0- Define an essential singularity
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A singularity z_0 where the Laurent series of f has infinitely many nonzero terms with negative powers- What is the Laurent series expansion of a function around z_0?
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Sum from n = -infinity to infinity of a_n(z-z_0)^n- Define the residue of f at an isolated singularity z_0
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The coefficient a_{-1} in the Laurent series expansion of f around z_0- State the Residue Theorem
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If f is analytic except for isolated singularities inside a closed contour C, then integral of f dz over C = 2*pi*i * (sum of residues inside C)- How do you compute the residue at a simple pole z_0?
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Res(f, z_0) = lim(z->z_0) [(z - z_0)*f(z)]
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