Chapter21 (2)

Each branch point serves as the origin of a polar coordinate system.

The differential equation is L d²i/dt² + Ri + 1/C ∫i dt = ωEo cos(ωt).

To show this, for each ε > 0, we need to find a δ > 0 such that |(z² + iz) - (-2)| < ε for all z’s in 0 < |z - z₀| < δ.

Compute r and θ, then use the formula with θ = 0 to obtain the first root, and the other n - 1 roots will be equally spaced on a circle of radius r^(1/n) centered at the origin.

Branch cuts are used to render multi-valued functions single-valued by restricting the argument to a specific interval.

If the limiting value obtained from one side is different from the limiting value obtained from the other side, then f'(z0) does not exist.

The complex function method is commonly used for obtaining the steady-state response.

f'(z) = 2z.

The angle φ1 increases, while φ2 increases and then decreases, equaling 0 at point b.

The homogeneous solution will inevitably tend to zero as t approaches infinity.

The function f(z) is continuous at z₀ if lim (z → z₀) f(z) = L and f(z₀) = L.

ip(t) = ωEoC Re[((1 - LCω²) + iRCω) e^(iωt)].

It is necessary that the Cauchy-Riemann equations be satisfied at that point.

Differentiability implies continuity, meaning that if a function is differentiable at a point, it must also be continuous at that point.

Functions that are analytic everywhere are called entire functions.

By considering the simpler equation L d²v/dt² + R dv/dt + 1/C v = ωEo e^(iωt) and seeking a solution of the form v_p(t) = Ae^(iωt).

Augustin-Louis Cauchy, Georg Friedrich Bernhard Riemann, and Karl Weierstrass.

The triangle inequality allows us to estimate the distance |z² + iz + 2| and ensure it remains less than ε.

At z = 0, the function 1/z has a singular point because the derivative fails to exist.

The limit of the difference quotient must be independent of the path of approach.

You obtain three values that are equally spaced on the circle of radius 6.538, with one example being -5.544 - 3.465i.

We must adopt the positive square root in the Pythagorean formula to ensure that the length of the hypotenuse is a unique, positive value.

It means that f(z) is differentiable at z0 and throughout some neighborhood of z0.

-e^(1-ia)/(1 - {a)

Functions like cosz, sinh z, cosh z, and polynomial functions of z are considered entire functions.

The derivative of a complex function f at a point g is defined as f'(g) = lim (f(z) - f(g)) / (z - g) as z approaches g, provided the limit exists.

The phase angle φ is given by φ = tan⁻¹(-R/(LCω - 1)).

The function f(z) = z is not differentiable anywhere.

The multi-valued nature of the logarithm can be illustrated by considering different branch cuts, which yield different values for the logarithm at the same point in the complex plane.

It takes n distinct values for various choices of k.

Virtually the entire body of complex variable theory deals only with functions of a single complex variable.

Yes, the function f(z) = e^z is entire.

f' can be represented by the expressions: F = g + vy = vy - iy = g - uy = vy + v.

L[Re v] + iL[Im v].

Applying a branch cut to the logarithm function makes it single-valued by defining the argument within a specific range, such as restricting the argument to values between -π and π.

The unique value is given by r^(1/n) * exp(i(θ + 2kπ)/n) where k = 0, ±1, ±2,...

Specifying an integer k in the logarithm function restricts its argument, effectively making the function single-valued.

Cubing f1 cubes its modulus and triples its argument.

In electrical engineering terminology, z is called the complex impedance of the circuit.

Graphs can be obtained using Maple by accessing the conformal command with specific input parameters.

It is sufficient that the Cauchy-Riemann equations be satisfied at the point and that the functions u and v are C1 in some neighborhood of that point.

C1 cos(wt) + C2 sin(wt).

ln denotes the ordinary logarithmic function of real variable theory, while log denotes the logarithmic function of a complex variable.

They are used to establish conditions for a function to be analytic by relating the partial derivatives of the real and imaginary components of the function.

One can find one root by inspection, then determine the others based on their equal spacing on the circle of radius 2.

The principal value of log z is defined by log z = ln r + iθ, where 0 < r < ∞ and -π < θ < π.

The derivative of f(z) = sinz is f'(z) = cosz.

Different paths can yield different values for φ1 and φ2, demonstrating the multi-valued nature of the function.

These inequalities help in keeping the function values f(z) arbitrarily close to L when z is sufficiently close to z₀.

The function 1/z is continuous for all z except z=0 because lim (z->0) 1/z does not exist.

Cauchy laid the groundwork of complex differential and integral calculus.

Yes, a function can be differentiable at a point but fail to be differentiable in some neighborhood of that point.

The polar form of complex numbers is essential for the calculation of powers of z and log z.

u(z, y) and v(z, y) must satisfy the Cauchy-Riemann equations.

The functions 2^n (n=0,1,2,...), e^z, sin(z), cos(z), sinh(z), and cosh(z) are continuous everywhere in the z plane.

Express z in polar form as log z = ln(r) + i(θ + 2πk) for k = 0, ±1, ±2,...

By making a legitimate trip from the point where θ = 0 to z = 2, keeping track of θ, the value is found to be 4.

If k is a rational number m/n, it is n-valued.

The Cauchy-Riemann equations are given as u_x = v_y and u_y = -v_x.

The finite cut renders f single-valued, although φ1 and φ2 remain multi-valued.

The mean value theorem can be used to express u and v in terms of their values at nearby points.

The limit must exist uniquely, independent of the way in which z approaches zp.

The nth roots fall on a circle of radius |z|^(1/n), centered at the origin, and are equally spaced on that circle.

The evaluation gives a different value than the one obtained from the bottom of the cut, demonstrating the impact of the branch cut.

Yes, a cut of finite length can sometimes be used, and the polar angle need not be limited to a 2π interval.

f(z) = 1/z is not differentiable at z = 0 because it is not continuous there.

They are necessary for differentiability but may not be sufficient.

The function can be expressed as log(z - a) = log(¢) = log(pe^iθ) = ln(p) + iθ, where ¢ is the vector from a to z.

The function f(z) = z^2 is differentiable only at z = 0 and is analytic nowhere.

If f(z) = u(z,y) + iv(z,y) is analytic in some domain D, then the partial derivatives of u and v exist and are continuous in D, leading to the Laplace equation.

Finite cuts allow encirclements of branch points, but force encirclements of both points, leading to cancellation of sign changes.

L[Re v] = F cos(wt) and L[Im v] = F sin(wt).

Each choice of k produces a distinct value of log z, making it infinite-valued.

If k is irrational, z^k is infinite-valued.

The integral can be evaluated by applying integration by parts or by recognizing that I = ∫ e^(-cos az) dz can be expressed in terms of the real part of a complex exponential, leading to simplifications.

f(z) can be expressed in polar coordinates as J(z) = u(r, θ) + iv(r, θ).

As Az approaches 0, Q1 and Q2 approach Py, allowing for the evaluation of limits in the expression for f.

The branch cut is introduced to prevent encirclements of θ = 0, thereby avoiding multi-valuedness.

Squaring f1 squares its modulus and doubles its argument.

The values of Fy simply repeat the values Fy1, Fy2, Fy3 over and over.

If the limit of the difference quotient does not exist uniquely, the function is not differentiable at that point.

A branch point occurs at z = a, as encircling this point causes multi-valuedness in the function.

The choice of branch cut is dictated by the context of the physical application; in the absence of context, any branch cut is as good as another.

The cut must emanate from the branch point and render the function single-valued in the cut plane.

A multi-valued function contradicts the definition of a function, which states that for a given input, there must be a unique output.

If a function is not analytic at a point z0, it is considered singular at that point.

The function has branch points at z = 2 and z = -2, each requiring its own branch cut.

Log z is analytic everywhere in the cut plane, which does not include the origin.

The function z^k is single-valued only if the exponent k is an integer.

Taking the derivative gives u_{xx} + u_{yy} = 0 in D, indicating that the functions satisfy the Laplace equation.

The functions u and v must be continuous in some neighborhood of the point where the equations are being evaluated.

sin z = (e^(iz) - e^(-iz)) / (2i), cos z = (e^(iz) + e^(-iz)) / 2, tan z = sin z / cos z, cot z = 1 / tan z, sec z = 1 / cos z, csc z = 1 / sin z.

If f(z) is a complex-valued function, then the integral can be expressed as: ∫f(z) dz = ∫[Re(f(z)) + i Im(f(z))] dz = ∫Re(f(z)) dx + i ∫Im(f(z)) dz.

Equations (25) and (26) can simplify the evaluation of certain integrals by separating the real and imaginary parts of a complex function during integration.

It allows for a particular solution v(t) to be found in the form Ae^(iwt), which is simpler than the two-term form C1 cos(wt) + C2 sin(wt).