The Radius of Convergence

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Consider the next power series.

$$\sum_{n=0}^∞ a_nz^n･･･(1)$$

We take the absolute value of \(a_n\) and \(z\) which make up this power series.

$$\sum_{n=0}^∞ |a_n||z|^n･･･(2)$$

This power series (2) is called absolute convergence if it is finite. Then if the power series (2) converges, the power series (1) also converges.

The convergence radius R refers to the boundary between the area the power series (2) converges absolutely and doesn’t. If \(|z|<R\), the power series (1) converges absolutely. Then if \(|z|>R\), it diverges. When \(|z|=R\), some power series converge, others diverge. Therefore, we need to think about converging or diverging for each power series.

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Laplace Transform and Algebraic Equation

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Laplace transform refers to the following conversion \(F(s)\).

$$F(s)=\int_0^∞ f(t)e^{-st}dt$$

The function before Laplace transform \(f(t)\) depends on \(t\), but the function after Laplace transform \(F(s)\) depends on \(s\).

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What Is Quantum Computers?

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On 20th November 2017, NTT announced a system using domestic quantum computer for general people. Recently we often hear about quantum computers, but what is the difference between such computers and ordinary computers?

In this article, I will briefly touch on the difference between quantum computer and conventional computer, and a quantum dot used for quantum computer.

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How to Obtain Eigenvalues and Eigenvectors

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The eigenvalue \(λ\) and the eigenvector \(φ\) of the matrix \({\bf A}\) satisfy the following equation.

$${\bf A}φ=λφ$$

Let’s see how to find this eigenvalue \(λ\) and eigenvector \(φ\).

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Characteristics of Coil, Resistor and Capacitor

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At university we often think of series RLC circuits. L is a coil, R is a resistance, and C is a capacitor. The relationship between the voltage applied to each electronic component and the current is given as follows.

Voltage of Coil \(V_L\)

$$V_{L}(t)=L\frac{dI(t)}{dt}$$

Voltage of Resistance \(V_R\)

$$V_{R}(t)=RI(t)$$

Voltage of Capacitor \(V_C\)

$$V_{C}(t)=\frac{1}{C} \int I(t) dt=\frac{Q(t)}{C}$$

\(L\):Self-inductance of the coil　\(R\):Resistance   \(C\):Capacitance   \(Q(t)\):Charge stored in the capacitor

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Centrifugal Force and Centripetal Force

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In order to understand the centrifugal force and the centripetal force, it is first necessary to understand the imaginary force that is related with observer’s condition.

The centripetal force is the real force required for the circular movement of the object. On the other hand, the centrifugal force is the imaginary force that works when the observer moves with the object.

The direction of the centripetal force and the centrifugal force are diametrically opposed, and their magnitudes are the same. Therefore, when the observer moves with the object, it seems that the centrifugal force and the centripetal force are balanced.

The magnitude of the centripetal force or the centrifugal force \(F\) is

$$F=mrω^2=m\frac{v^{2}}{r}.$$

\(m\):mass   \(r\):radius of the orbit of the object   \(ω\):angular velocity[rad/s]　\(v\):object speed[m/s]

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Cauchy-Riemann Equation and Holomorphic Function

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A complex-valued function \(f(z)\) is said to be a holomorphic function if it is differentiable at every point in its domain.

By the way, let \(f(z)\) be defined as

$$f(z)=u(x,y)+iv(x,y),$$

where \(z\) is a complex variable, \(u(x,y)\) and \(v(x,y)\) are real-number functions, and \(x\) and \(y\) are real variables. The following two equations are collectively called Cauchy-Riemann’s equations.

$$\frac{∂u(x,y)}{∂x}=\frac{∂v(x,y)}{∂y}$$

$$\frac{∂u(x,y)}{∂y}=-\frac{∂v(x,y)}{∂x}$$

You can determine the complex-valued function \(f(z)\) is a holomorphic function when \(u(x, y)\) and \(v(x, y)\) satisfy the Cauchy-Riemann equations above.

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Differential Form and Integral Form of Maxwell Equation

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Differential Form of Maxwell Equation

Gauss’s law

$$div {\bf D}({\bf r},t)=ρ({\bf r},t)$$

Law of the magnetic flux conservation

$$div {\bf B}({\bf r},t)=0$$

Maxwell-Ampere law

$$rot {\bf H}({\bf r},t)={\bf i}({\bf r},t)+\frac{∂{\bf D}({\bf r},t)}{∂t}$$

Faraday’s law of induction

$$rot {\bf E}({\bf r},t)=-\frac{∂{\bf B}({\bf r},t)}{∂t}$$

Relation between electric field and electric flux density, magnetic field and magnetic flux density

$${\bf D}({\bf r},t)=ε{\bf E}({\bf r},t)$$
$${\bf B}({\bf r},t)=μ{\bf H}({\bf r},t)$$

\({\bf E}({\bf r},t)\):Electric field   \({\bf D}({\bf r},t)\):Electric flux density   \({\bf H}({\bf r},t)\):Magnetic field   \({\bf B}({\bf r},t)\):Magnetic flux density

\(ρ({\bf r},t)\):Charge density   \({\bf i}({\bf r},t)\):Current density   \(ε\):Dielectric constant (Dielectric)   \(μ\):Magnetic permeability (Permeability)

The first four equations are collectively called the Maxwell equation. The form is called the differential form of the Maxwell equation. The latter two supplements wrote the electric field and electric flux density, and the relational expressions of the magnetic field and the magnetic flux density, respectively.

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Fourier Transform

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I thought about Fourier series expansion before, but this time it is an introduction of Fourier transform.

Fourier transform

$$F(k)=\int_{-∞}^{∞} f(x)e^{-ikx} dx$$

inverse Fourier transform

$$f(x)=\frac{1}{2π}\int_{-∞}^{∞} F(k)e^{ikx} dk$$

Suppose a wave is given by the function \(f(x)\). Using Fourier transform, the function of wave \(f (x)\) is expressed as a function \(F(k)\) of wavenumber \(k\). \(F(k)\) is the function which decomposes \(f(x)\) into the sine waves of all wavenumbers \(k\) and expresses the size of each the sine wave as a function. You can convert the wave \(f (x)\) to \(F(k)\) by Fourier transform and \ (F (k) \) to \(F(k)\) by inverse Fourier transform.

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