sound-waves

Principals of Modern Physics: 
Historical Survey
The term modern physics generally refers to the studies of those facts and theories, that concern the ultimate structure and interactions of matter, space and time. The three main branches of classical physics – mechanics, heat and electromagnetism — were developed over a period of approximately two centuries prior to 1900.

Newton’s mechanics dealt successfully with the motions of bodies of macroscopic size moving with low speeds, and provided a foundation form many of the engineering accomplishments of the eighteenth and nineteenth centuries. With Maxwell’s discovery of the displacement current and the completed set of electromagnetic field equations, classical technology received new impetus.

Galilean principle of relativity, recognized by Newton, the laws of mechanics should be expressed in the same mathematical constant velocity relative to each other. The transformation equations, relations measurements in two relatively moving inertial frames, were not consistent with the transformations obtained by Lorentz from similar considerations of form-invariance applied to Maxwell’s equations. 
The first major step toward a deeper understanding of the nature of space and time measurements was due to Albert Einstein, whose special theory of relativity resolved this inconsistency, between mechanics and electromagnetism by showing, among other things, that Newtonian mechanics is only a first approximation to a more general set of mechanical laws; the approximation is, however, extremely good when the bodies move with speeds which are small compared to the speed of light. Among the important results obtained by Einstein, was he equivalence of mass and energy, expressed in the famous equations E=mcsquared.

From a logical standpoint, special relativity lies at the heart of modern physics. The hypothesis that electromagnetic radiation energy is quantized in bunches of amount h,v, where v is the frequency and h is a constant, enabled Planck to explain the intensity distribution of black-body radiation.
Einstein also applied the quantum hypothesis to photons in an explanation of the photoelectric effect. This hypothesis was found to be consistent with special relativity. Similarly, Bohr’s postulate – that the electrons angular momentum in the hydrogen atom is quantized in discrete amounts 0 enabled him to explain the positions of the spectral lines in hydrogen.
Ad hoc quantization rules
Lois de Broglie proposed on the basis of relativity theory, that waves were associated with material particles, that the foundations of a correct quantum theory were laid. Schrodinger in 1926 proposed a wave equation describing the propagation of these particle waves, and developed a quantitative explanation of atomic spectral line intensities. In a few years thereafter, the success of the new wave mechanics revolutionized physics.

Sommerfeld to explain the behavior of electrons in a metal. Bloch’s treatment of electron waves in crystals simplified the application of quantum theory to problems of electrons in solids. Dirac, discovered that a positively charged electron should exist; this particle, called a positron was later discovered.
Modern physics has steadily progressed toward the study of smaller and smaller features of the microscopic structure of matter, using the conceptual tools of relativity and quantum theory.
Schrodinger
Starting in 1932 with the discovery of the neutron by Chadwick
Nuclear fission and nuclear fusion are byproducts of these studies, which are still extremely active.
Over fifty of the so-called elementary particles have been discovered. these particles are ordinarily created by collisions between high-energy particles of some other type, usually nuclei or electrons.

It should be emphasized that one of the most important unifying concepts in modern physics is that of energy.
The famous variational principles of Hamilton and Lagrange expressed Newtonian laws in a different form, by woking with mathematical expressions for the kinetic and potential energy of a system. Einstein showed that energy and momentum are closely related in relativistic transformation equations, and established the equivalence of energy and mass.
De Broglie’s quantum relations….
Schrodingers wave equation is
the most sophisticated expressions of modem-day relativistic quantum theory are variational principles, which involve the energy of a system expressed in quantum=mechanical form.
Quantum systems are states of definite energy.
Another very important concept used throughout modern physics is that of probability.

We shall begin in Chapter 2 with a brief introduction to the concept of probability and the rules for combining probabilities. This material will be used extensively in later chapters on the quantum theory and on statistical mechanics.

Notation and Units
The well-known meter-kilogram-second (MKS) system of units will be used in this book. Vetors will be denoted by boldface type, such as F for force. In these units, the force on a point charge of O which is that of coulombs, moving with velocity v in meters per second, at a point where the electric field in E volts per meter and the magnetic field is B webers per square meter, is the Lorentz force;
F=Q(E+v x B)
F=Q(E+v x B)
F= Q (E+V x B)

where v x B denotes the actor cross-product of v and B. The potential in volts produced by a point change Q at a distance r from the position of the charge is given by Coulumb’s law:
V(r) = Q/4pi(e)0 = 9 x 10 to the ninthnewtons-msquared.coulomb squared
These particular expressions form electromagnetic theory are mentioned here because they will be used in subsequent chapters.
Units of energy and momentum
While in the MKS system of units the basic energy unit is the joule, in atomic and nuclear physics several other units of energy have fond widespread use.
The electron volt is defined as the amount of work done u0pon an electron as it moves through a potential difference of one volt.
thus
the electron volt is an amount of energy in joules equal to the numerical value of the electron’s charge in columbs. To convert energies from joules toeV, or from eV to joules, one divides or multiplies by e, respectively.
For examples, for a particle with the mass of the electron, moving with a speed of 1 percent of the speed of light, the kinetic energy would be

In nuclear physics most energies are of the order of several million electron volts, leading to the definition of a unit called the MeV.
Then if the momentum p in MeV/c is known, the energy in MeV is numerically equal to p. Thus, in general, for photons.
E(inMeV/c)

Atomic Mass Unit

The atomic mass unit, abbreviated amu, is chosen in such a way that the mass of the most common atom of carbon, containing six protons, and six neutrons in a nucleus surrounded by six electrons is exactly 12.000000

In addition, a slightly different choice of atomic mass unit is commonly used in chemistry. All atomic masses appearing in this book are based on the physical scale, using carbon as the standard.
The conversion from amu on the physical scale to kilograms may be obtained by using the fact that one gram-molecular weight of a substance contains Avogadro’s number
Thus, exactly 12.00000

Propagation of waves, Phase and group speeds
Broglie’s probability waves of quantum theory, lattice vibrations in solids, light waves, and so on. these waves motions can be described by a displacement, or amplitude of vibration of some physical quantity, of the form.

Where À and Ø are constants and where the wavelength and frequency of the wave are given by
Here the angular frequency is denoted by w=w(k), to indicate that the frequency is determined by wavelength, or wavenumber k. This frequency wavelength relations, w = w(k) is called a dispersion relation and arises because of the basic physical laws satisfied by the particular wave phenomenon under investigation. For example, for sound waves in air, Newton’s second law of motion and the adiabatic gas law imply that the dispersion relation in w =vk.

Where v is a constant
This represents a wave propagating in the positive x direction

w = w/k
In nearly all cases, the wave phenomena which we shall discuss obey the principle of supposition – namely, that if waves from two or more sources arrive at the same physical point, then the net displacement is simply the sum of the displacements from the individual waves. 

Reinforcement of destructive interference can then occur as one wave gains on another of different wavelength. The speed with which the regions of constructive or destructive interference advance is known as the group speed.
To calculate this speed, consider two trains of waves of the form of Equation, of the same amplitude but of slightly different wavelength and frequency such as

This expression represents a wave traveling with phase speed w/k, and with an amplitude given by

The amplitude is a cosine curve, the spatial distance between two successive zeros of this curve at a given instant is pi.triangle k and is the distance between two successive regions of destructive interfrence. These regions propagate with the group speed v subset g, given by

and the phase and group speeds are equal. On the other hand, for surface gravity waves in a deep sea, the dispersion relations is
where g is the gravitation acceleration. T is the surface tension and p is the density. Then the phase speed is

If the phase speed is a decreasing function of km or an increasing function of wavelength, the the phase speed is greater than the group speed, and individual crests within a region of construvitve interference – i.e. within a group of waves – travel from rear to front, crests disappearing at the front and reappearing at the rear of the group.

Complex numbers
Because the use of complex numbers is essential in the discussions of the wavelike character of particles, a brief review of the elementary properties of complex numbers is given here. A complex number is of the form trident = a+ib, where a and b are real numbers and i is the imaginary unit, i squared = -1. the real part of the trident, is a, and the imaginary part is b:

A complex number trident – a +iB can be represented as a vector in two dimensions, with the x component of the sector identified with Re trident and the y component

We can calculate the magnitude of the square of the vector by multiplying trident by its complex conjugate

The complex exponential function, e to the iØ whee ø is a real function or number, is a particular importance; this function may be defined by the power series

Then, replacing i squared everywhere that it appears by -1 and collecting real and imaginary terms, we find that

Moivre’s theroum

The integral of an exponential function of the form e to the c to the x is

and this is also valid when is complex: For example,
The complex exponential function is a periodic function with 2pi Thus
More generally, if n is any positive integer or negative integer.
Introduction

Historical Survey
The term modern physics generally refers to the study of those facts and theories that concern the ultimate structure and interactions of matter, space and time. The tree main branches of classical physics – mechanic, heat and electromagnetism . Newton’s mechanics dealt successfully with the motions of bodies of macroscopic size moving with low speeds, and provided a foundation for many of the engineering accomplishments of the nineteenth century. With Maxwell’s discovery of the displacement current and the completed set of electromagnetic field equations, classical technology received new impetus.

Galilean principle of relativity, recognized by Newton, the laws of mechanics should be expressed in the same mathematical form by observers in different inertial frames of reference, which are moving with constant velocity relative to each other. The transformation equations relating measurements in two relatively moving inertial frames, were not consistent with the transformations obtained by Lorentz from similar considerations of form-invariance applied to Maxwell’s equations.
The fist major step toward a deeper understanding of the nature of space and time measurements was due to Albert Einsten, whose special theory of relativity, resolved the inconsistency between mechanics and electromagnetism by showing, among other things, that Newtonian mechanics is only a fist approximation to a more general set of mechanical laws’ the approximation is however, extremely good when the bodies move with speeds which are small compared to the speed of light.
Among the important results obtained by Einstein was the equivalence of mass and energy, expressed in the famous equations, E= MCsquared.
From a logical standpoint, special relativity lies at the heart of modern physics. The hypothesis that electromagnetic radiation energy is quantized in bunches of amount hv, where v is the frequency and h is a constant, enabled Plank to explain the intensity distribution of black-body radiation. 
Einstein also applied the quantum hypothesis to photons in an explanation of the photoelectric effect. This hypothesis was found to be consistent with special relativity. Similarly, Bohr’s postulate – that the electron’s angular momentum in the hydrogen atom is quantized in discrete amounts – enabled him to explain the positions of the special lines of hydrogen.
Ad hoc quantizations rules
1924, when Louis de Broglie proposed, that waves were associated with material particles, that the foundations of a correct quantum theory were laid. 
Schrodinger in 1926 proposed a wave equation describing the propagation of these particle waves, and developed a quantitative explanation of atomic spectral line intensities. In a few years thereafter, the success of the new wave mechanics revolutionized physics.
Pauli’s expulsion principle
Sommerfeld to explain the behavior of electrons in a metal.
Bloch’s treatment of electron waves in crystals simplified the application of quantum theory to problems of electrons in solids. 
Dirac discovered that a positively charged electron should exist; this particle, called a positron was later discovered.
Modern physics has steadily progressed toward the study of smaller and smaller features of the microscopic structure of matter, using the conceptual tools of relativity and quantum theory
Scrodinger
1932, with the discovery of the neutron by Chadwick
Nuclear fission and nuclear fusion are byproducts of these studies, which are still extremely active.

Over fifty of the so called elementary particles have been discovered. these particles are ordinarily created by collisions between high energy particles of some other type, usually nuclei or electrons.

It should be emphasized that one of the most important unifying concepts in modern physics is that of energy.
The famous variational principles of Hamilton and Lagrange expressed Newtonian laws in a different form, by working with mathematical expressions for the kinetic and potential energy of a system. 
That energy and moments are closely related in relativistic transformation equations, and established the equivalence of energy and mass. De Brogile’s quantum relations
Schrodinger’s wave equation
The most sophisticated expressions of modern day relativistic quantum theory are variational principles, which involve the energy of a system expressed in quantum-mechanical form.

Quantum systems ares states of definite energy.
Another very important concept used throughout modern physics is that of probability.
Instead, they had to be described in terms of probabilities. It is somewhat ironic that probability was first introduced into quantum theory by Einstein in connection with the discovery of stimulated emission. Heisenberg’s uncertainty principle, and the probability interpretation of the Schrodinger wavefunction.
We shall begin in chapter 2 with a brief introduction to the concept of probability and the rules for combining probabilities. this material will be used extensively in later chapters on the quantum theory and on statistical mechanics.

Notion and Units

The well-known meter-kilogram-second (MKS) system of units will be used in this book. Vectors will be denoted by boldface type, such as F for force. In the units, the force on a point charge of O coulombs, moving with velocity v in meters per second, at a point where the electric field is E volts per meter and the magnetic field is B webers per square meter, is the Lorentz force; where vxB denotes the vector cross-product of v and B. The potential in volts produced by a point charge Q at a a distance r from the position of the charge is given by Coulomb’s law:
F=Q(E+vxB)
F=Q(E+vxB)
F=Q(E+vxB)
V(r) = Q/4piesubseto)
These particular expressions from electromagnetic theory are mentioned here because they will be used in subsequent chapters.
Boltzman’s constant

Units of Energy and Momentum
While in the MKS system of units the basic energy unit is the joule, in atomic and nuclear physics several other units of energy have found widespread use.
the electron volt is defined as the amount of work done upon an electron as it moves through a potential difference of one volt.

The electron is an amount of energy in joules equal to the numerical value of the electron’s charge in coulombs. To convert energies from joules to eV, or from Ev to Joules, one dives or multiples by e, respectively.

In nuclear physics most energies are of the order of several million electron volts, leading to the definition of a unit called MeV:

Then if the momentum p in MeV/c is known, the energy in MeV is numerically equal to to p. Thus, in general, for photos.
Atomic Mass Unti
The atomic mass unit, abbreviated amu, is chosen in such a way that the mass of the most common atom of carbon, containing six protons and six neutrons in a nucleus surrounded by six electrons, is exactly 12.000000amu.
In addition, a slightly different choice of atomic mass unit is commonly used i n chemistry. All atomic masses appearing in this book are based on the physical scale to kilograms may be obtained by using the fact that one gram-molecular weight of a substance contains Avogardo’s number, N subset ô = 6.022X 10 to the twenty-third. Thus, exactly 12.000

Propagation of Waves; Phase and Group Speeds

Broglie probability waves of quantum theory, lattice vibrations in solids, light waves, and so on. These wave motions can be described by a displacement, or amplitude of vibration of some physical quantity, of the form:
Where A and Ø are constants, and whee the wavelength and frequency of the wave are given by
here the angular frequency is denoted by w=w(K) to indicate that the frequency is determined by the wavelength, or wavenumber k. This frequency wavelength relation, w =w(k) is called a dispersion relation and arises because of the basic physical laws satisfied by the particular wave phenomenon under investigation. For example, for sound wavs in air, Newton’s second law of motion and the adiabatic gas law imply that the dispersion relation is w =vk
Where v is a constant

This represents a wave propagating in the positive x direction. W =w/k
In nearly all case, the wave phenomena which we shall discuss obey the principle of superposition – namely, that if wavs from two or more sources arrive at the same physical point, then the net displacement is simply the sum of the displacements from the individual waves. ….Reinforcement or destructive interference can then occur as one wave gains on another of different wavelength. The speed with which the regions of constructive or destruvtive interference advance in known as the group speed.
To calculate this speed, consider two trains of waves of the form of Equation of the same amplitude but of slightly different wavelength and frequency such as the following equation
This expressions represents a wave travelings with phase speed w/k, and with an amplitude given by

the amplitude is a cosine curve; the spatial distance between two successive zeros of this curve at a given instant is pi /triangle k, and is the distance between two successive regions of destructive interference. These regions propagate with the group speed v subset g given by

And the case and group speeds are equal. On the other hand, for surface gravity waves in a deep sea, the dispersion relation is
where g is the gravitational acceleration. T is the surface tension and p is the density. Thus the phase speed is

If the phase speed is decreasing function of k, or an increasing function on wavelength, then the phase speed is greater than the group speed, and individual crests within a region of constructive interference – i.e. within a group of waves — travel from rear to front, crests disappearing fat the front and reappearing at the rear of the group.

Complex Number
Because the use of complex numbers is essential in the discussions of the wavelike character of particles, a brief review of the elementary properties of complex number is given here…..where a and b are real number and I is the imaginary unit, i squared = -1.. The real part of trident is a, and the i imaginary part is b

A complex number trident = a=ib, can be represented as a vector in two dimensions, w with the x component of the vector identified with Re trident, and the y component

of the vector identified with Im trident as in Figure 1.1. The square of the magnitude of the vector is

The complex conjugate of trident = a +ib is denoted by the symbol trident* and is obtained by replacing the imaginary but b – i

We can calculate the magnitude of the square of the vector by multiplying trident by its complex conjugate
The complex exponential function, e to the iø where ø is a real function or number, is a particular importance; this function may be defined by the power series

Then replacing i squared everywhere that is appears by -1 and collecting real and imaginary terms, we find that

Mivre’s theorem:

The integral of an exponential function of the form e to the cx is
and this is also valid when c is complex. For example,
The complex exponential function is a period function with period 2 pi, thus, and more general, if n is any positive integer or negative integer.

Introduction
Historical Survey

The term modern physics refers to those facts and theories developed, that concern the ultimate structure and interactions of matter, space and time. The three main branches of classical physics – mechanics, heat and electromagnetism.
Newton’s mechanics dealt successfully with the motions of bodies of macroscopic size moving with low speeds, and provided a foundation for many of the engineering accomplishments of the eighteenth and nineteenth centuries. Maxwell’s discovery of the displacement current and the completed set of electromagnetic field equations, classical technology received new impetus: 
Galilean principle of relativity, recognized by Newton, the laws of mechanics should be expressed in the same mathematical form by observers in different inertial frames of reference, which are moving with the constant velocity relative to each other. The transformation equations, relating measurements in two relatively moving inertial frames, were not consistent with the transformation obtained by Lorentz from similar considerations of form invariance applied to Maxwell’s equation. 

The first major step toward a deeper understanding of the nature fo space and time measurements was due to Albert Einstein, whose special theory of relativity resolved one inconsistency between mechanics and electromagnetism by showing, among other things, that Newtonian mechanics is only a first approximation to a more general set of mechanical laws; the approximation is, however, extremely good when the bodies move with speeds which are small compared to the sped of light.Among the important results obtained by Einstein was the equivalence of mass and energy, expressed in the famous equation E = mc squared.
Planck to explain the intensity distribution of black-body radiation. 
The quantum hypothesis to photons is an explanation of the photoelectric effect. The hypothesis was found to be consistent with special relativity. Similarly, Bohr’s postulate – that the electron’s angular momentum in the hydrogen atom is quantized in discrete amounts – enabled hi to explain the positions of thee spectral lines in hydrogen.

Ad hoc quantization rules

1924, when Louis de Broglie proposed on the basis of relativity theory, that waves were associated with material particles, that the foundations of a correct quantum theory were laid.
Schrodinger in 1926 proposed a wave equation describing the propagation of these particle waves, and developed a quantitative explanation of atomic spectral one intensities. In a few years thereafter, the success of the new wave mechanics revolutionized physics.

Pauli’s exclusion principe

Sommerfeld to explain the behavior of electrons in metal. Bloch’s treatment of electron waves in crystals simplified the application of quantum theory to problems of electrons in solids.
Dirac, discovered that a positively charged electron should exist; this particle, called a positron, was later discovered.
Modern physics has steadily progressed toward the study of smaller and smaller features of the microscopic structure of matter, using the conceptual tools of relativity and quantum theory.

Schrodinger’s

1932 with the discovery of the neutron by Chadwick

Nuclear fission and nuclear fusion are byproducts of these studies, which are still extremely active.

Over fifty of the so called elementary particles have been discovered. These particles are ordinarily created by collisions between high-energy particles of some other type, usually nuclei or electrons.

It should be emphasized that one of the most important unifying concepts in modern physics is that of energy.

The famous variational principles of Hamilton an Lagrange expressed Newtonian laws in a different form, by working with mathematical expressions for the kinetic and potential energy of a system. 
Energy and momentum are closely related in relativistic transformation equations, and established the equivalence of energy and mass. De Broglie’s quantum relations

Schrodinger’s wave equation
The most sophisticated expressions of modern-day quantum theory are variational principles, which involve the energy of a system expressed in quantum-mechanical form.
Quantum systems are states of definite energy.
Another very important concept used throughout modern physics is that of probability.
Instead, they had to be described in terms of probabilities. It is somewhat ironic that probability was first introduced into quantum theory boy Einstein in connection with his discovery of stimulation emission.. Heisenberg’s uncertainty principle, and the probability interpretation of the Schrodinger wavefunction

we shall begin in chapter 2 with a brief introduction to the concept of probability and to the rules for combining probabilities. This material will be used extensively in later chapters in the quantum theory and on statistical mechanics.
Notation and Unit

The well-known meter-kilogram-second (MKS) system of units will be used in this book. Vectors will be denoted by boldface type, such as F for force. In these units, the force on a point charge of Q (coulombs) moving with velocity v in meters per secong, at a point where the electric field is E volts per meter and the magnetic field is B webers per square meter, is the Lorentz force;
F=Q (E+VxB)

Where vxv denotes the vector cross-product of v and B. The potential in volts produced by a point charge Q at a distance r from the position of the charge is given by Coulomb’s law
Where the constant e subset o is given by

These particular expressions from electromagnetic theory are mentioned here because they will be used in subsequent chapters.

Boltzman’s constant

Units of Energy and Momentum
While in the MKS system of units the basic energy unit is the joule, in atomic and nuclear physics several other units of energy have found widespread use.
The electron volt is defined as the amount of work done upon an electron as it moves through a potential difference of one volt
The electron volt is an amount fo energy in joules equal to the numerical value of the electron’s charge in coulombs. To covert energies from joules to eV, or from eV to joules, one divides or multiplies by e, respectively

In nuclear physics most energies are of the order of several million electron volts, leading to the definition of a unit called MeV:

Then if the momentum p in MeV/c is known, the energy is MeV is numerically equal to p. Thus, in general, for photons
Atomic Mass Unit
The atomic mass unit, abbreviated amu, is chosen in such a way that the mass of the most common atom of carbon, containing six protons and six neutrons in a nucleus surround by six electrons, is exactly 12.0000000amu

In addition, a slightly different choice of atomic mass unit is commonly used in chemistry. All atomic masses appearing in this book are based on the physical sale, using carbon as the standard.
The conversion from amu on the physical scale to kilograms may be obtained by using the fact that one gram-molecular weight of a substance contains Avogardo’s number, Nsubset ) = 6.022 X 10 to the 23 of molecules. Thus, exactly 12.00000amu, grams of C to the twelfth atoms contains N to the atoms and

Broglie’s probability waves of quantum theory, attire vibrations in solids, light waves, and so on. These wave motions can be described by a displacement, or amplitude of vibrations of some physical quality, of the form

Where A and Ø are constants, and where the wavelength and frequency of the wave are given by
Here the angular frequency is denoted by w=w(k) to indicate that the frequency is determined by the wavelength, or wavenumber k. This frequency wavelength relation, w – w (k) is called a dispersions relation and arises because of the basic physical laws satisfied by the particular wave phenomenon under investigation. For example, for sound waves in air, Newton’s second law of motion and adiabatic gas law imply that the dispersion relation is w=vk
Where v is a constant
This represents a wave propagating in the positive x direction
In nearly all cases, the wave phenomena which we shall discuss obey the principle of superposition – namely that if waves from two or more sources arrive at the same physical point, then the net displacement is simply the sum of the displacements from the individual waves.
Angular frequency w

Reinforcement or destructive interference can then occur as one wave gains on another of different wavelength. The speed with which the regions of the constructive or destructive interference advance is known as the group speed.
To calculate speed, consider two trains of waves of the from fo Equation of the same amplitude but of slightly different wavelength and frequency, such as
This expression represents a wave traveling with phase speed w/k and with an amplitude given by
The amplitude is a cosine curve; the spatial distance between two successive zeros, of this curve at a given instant is pi/triangle k and is the distance between two successive regions of destructive interference. These regions propagate with the group seed, v subset g given by

and the case and group speeds are equal. on the other hand, for surface gravity waves in a deep sea, the dispersion relation isWhere g is the gravitational acceleration. T is the surface tension and p is the density. Then the phase speed i s tractive interference – i.e.

If the phase speed is a decreasing function of k, , or an increasing function of wavelength, then the phase speed is greater than the group speed, and individual crests with a region of constructive interference – i.e. within a group of waves- travel from rear to front, crest disappearing at the front and reappearing at the rear of the group.
Complex number
Because the use of complex numbers is essential in the discussion so of the wavelike character of particles, a brief review of the elementary properties of complex numbers is given here. Trident – a +iB, where a and b are real number and i is the imaginary unit, i squared = – 1. The real part of trident, is a, and the imaginary part is b.
A complex number trident = a +ib, can be represented as a vector in two dimensions, with the x component of the vector identified with Re trident, and the y component

Of the vector identified with Im trident as in Figure 1.1. The square of the magnitude of the vector is
The complex conjugate of trident = a+ibsquared is denoted by the symbol trident * and is obtained by replacing the imaginary unit i by – i.
tident*=a+-ib
We can calculate the magnitude of the square of the vector by multiplying trident by its complex conjugate.
The complex exponential function e to the iø where ø is a real function or number, is a particular importance; this function may be defined by the power series
Then replacing i squared everywhere that it appears by -1 and collecting real and imaginary terms, we fin that
Moivre’s theorem
The integral of an exponential function of the from e to the cx is
and this is also valid when c is complex. For example: See notes
The complex exponential function is a period function with period 2 pi. Thus
More generally, if n is any positive integer or negative integer.

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