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harmonic series
A Wisdom Archive on harmonic series

harmonic series A selection of articles related to harmonic series

We recommend this article: harmonic series - 1, and also this: harmonic series - 2.

harmonic series

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ARTICLES RELATED TO harmonic series

harmonic series: Encyclopedia II - Harmonic number - Introduction

The generalized harmonic number of order n of m is given by . Note that n may be equal to , provided m > 1. And If , while , the harmonic series does not converge and hence the harmonic number does not exist. Other notations occasionally used include The special case of m = 1 is simply called a harmonic number and is frequently written w ...

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Harmonic number, Harmonic number - Introduction, Harmonic number - Applications, Harmonic number - Generalizations, Harmonic number - Generating functions

Read more here: » Harmonic number: Encyclopedia II - Harmonic number - Introduction

harmonic series: Encyclopedia II - Digamma function - Recurrence formula

The digamma function satisfies the recurrence relation Thus, it can be said to "telescope" 1/x, for one has where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula More generally, one has ...

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Digamma function, Digamma function - Relation to harmonic numbers, Digamma function - Integral representations, Digamma function - Taylor series, Digamma function - Newton series, Digamma function - Reflection formula, Digamma function - Recurrence formula, Digamma function - Special values

Read more here: » Digamma function: Encyclopedia II - Digamma function - Recurrence formula

harmonic series: Encyclopedia II - Digamma function - Newton series

The Newton series for the digamma follows from Euler's integral formula: where is the binomial coefficient. ...

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Digamma function, Digamma function - Relation to harmonic numbers, Digamma function - Integral representations, Digamma function - Taylor series, Digamma function - Newton series, Digamma function - Reflection formula, Digamma function - Recurrence formula, Digamma function - Special values

Read more here: » Digamma function: Encyclopedia II - Digamma function - Newton series

harmonic series: Encyclopedia II - Digamma function - Taylor series

The digamma has a rational zeta series, given by the Taylor series at z=1. This is , which converges for |z|<1. Here, ζ(n) is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function. ...

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Digamma function, Digamma function - Relation to harmonic numbers, Digamma function - Integral representations, Digamma function - Taylor series, Digamma function - Newton series, Digamma function - Reflection formula, Digamma function - Recurrence formula, Digamma function - Special values

Read more here: » Digamma function: Encyclopedia II - Digamma function - Taylor series

harmonic series: Encyclopedia II - Proof that the sum of the reciprocals of the primes diverges - First proof

Euler took the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the logarithm of each side, then he used the Taylor series expansion for ln(1 − x) as well as the sum of a geometric series: for some fixed constant C. Since the sum of the reciprocals of the first n positive integers is asymptotic to ln(n), (i.e. their rati ...

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Proof that the sum of the reciprocals of the primes diverges, Proof that the sum of the reciprocals of the primes diverges - The harmonic series, Proof that the sum of the reciprocals of the primes diverges - First proof, Proof that the sum of the reciprocals of the primes diverges - Second proof, Proof that the sum of the reciprocals of the primes diverges - Third proof, Proof that the sum of the reciprocals of the primes diverges - External link

Read more here: » Proof that the sum of the reciprocals of the primes diverges: Encyclopedia II - Proof that the sum of the reciprocals of the primes diverges - First proof

harmonic series: Encyclopedia II - Proof that the sum of the reciprocals of the primes diverges - Third proof

Here is another proof that actually gives an estimate for the sum; in particular, it shows that the sum grows at least as large as lnlnn. The proof is an adaptation of the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes. The proof rests upon the following facts: Every positive integer n can be expressed as the product ...

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Proof that the sum of the reciprocals of the primes diverges, Proof that the sum of the reciprocals of the primes diverges - The harmonic series, Proof that the sum of the reciprocals of the primes diverges - First proof, Proof that the sum of the reciprocals of the primes diverges - Second proof, Proof that the sum of the reciprocals of the primes diverges - Third proof, Proof that the sum of the reciprocals of the primes diverges - External link

Read more here: » Proof that the sum of the reciprocals of the primes diverges: Encyclopedia II - Proof that the sum of the reciprocals of the primes diverges - Third proof

harmonic series: Encyclopedia II - Khinchin's constant - Hölder means

The Khinchin constant can be viewed as the first in a series of the Hölder means of the terms of continued fractions. Given an arbitrary series {an}, the Hölder mean of order p of the series is given by When the {an} are the terms of a continued fraction expansion, the constants are given by This is obtained by taking the p-th mean in conjunction with the Gauss-Kuzmin distribution. The value for K0 ...

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Khinchin's constant, Khinchin's constant - Series expressions, Khinchin's constant - Hölder means, Khinchin's constant - Harmonic mean

Read more here: » Khinchin's constant: Encyclopedia II - Khinchin's constant - Hölder means

harmonic series: Encyclopedia II - Khinchin's constant - Series expressions

Khinchin's constant may be expressed as a rational zeta series in the form or, by peeling off terms in the series, where N is an integer, held fixed, and ζ(s,n) is the Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also ...

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Khinchin's constant, Khinchin's constant - Series expressions, Khinchin's constant - Hölder means, Khinchin's constant - Harmonic mean

Read more here: » Khinchin's constant: Encyclopedia II - Khinchin's constant - Series expressions

harmonic series: Encyclopedia II - Natural horn - Handhorn Technique

Before starting to learn handhorn technique get to know the harmonic series of the horn or the notes that can naturally be played without valves. For every harmonic there are a few things we can do to give us a near chromatic range: To play one half-step lower close off the bell a bit. This will take some experimenting to figure out how much but it should be relatively easy. To play one half-step higher fully stop the bell. For modern horns we are encouraged to make this sound quite nasally. For the natural horn ...

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Natural horn, Natural horn - Handhorn Technique, Natural horn - Natural Horn Repertoire, Natural horn - Natural Horn and The Modern Horn

Read more here: » Natural horn: Encyclopedia II - Natural horn - Handhorn Technique

harmonic series: Encyclopedia II - Fourier series - Definition of Fourier series

Suppose that f(x), a complex-valued function of a real variable, is periodic with period 2π, and is square-integrable over the interval from −π to π. Let Each Fn is called a Fourier coefficient. Then, the Fourier series representation of f(x) is given by Each term in this sum is called a Fourier mode or a harmonic. In the important special case of a real-valued function f< ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Definition of Fourier series

harmonic series: Encyclopedia II - Harmonic number - Generalizations

Euler's integral formula for the harmonic numbers follows from the integral identity which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series which is just the Newton's generalized binomial theorem. The interpolating function is in f ...

See also:

Harmonic number, Harmonic number - Introduction, Harmonic number - Applications, Harmonic number - Generalizations, Harmonic number - Generating functions

Read more here: » Harmonic number: Encyclopedia II - Harmonic number - Generalizations

harmonic series: Encyclopedia II - Multiphonic - How multiphonics work

In general, when playing a wind instrument, the tone that comes out consists of the fundamental—the pitch usually identified as the note being played—as well as pitches with frequencies that are integer multiples of the frequency of the fundamental. (Only pure sine wave tones lack these overtones.) Normally, we perceive only the fundamental pitch as being played. By controlling the air flow through the instrument and the shape of the column (by changing fingering or valve position), a player may produce two distinct tones not part of the same h ...

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Multiphonic, Multiphonic - Technique, Multiphonic - How multiphonics work, Multiphonic - Notation, Multiphonic - Use in literature

Read more here: » Multiphonic: Encyclopedia II - Multiphonic - How multiphonics work

harmonic series: Encyclopedia II - Multiphonic - Use in literature

The first real use of multiphonics in literature are of the brass "horn chord" style. Carl Maria von Weber used this technique in his horn literature; it was used in the 18th century. Woodwind multiphonics and brass lip multiphonics did not make appearances in classical music until the 20th century, with pioneering compositions such as Luciano Berio's Sequenzas for solo wind instruments using them extensively. However, usage in jazz predates this, having been used as early as the 1940s, with Illinois Jacquet an early proponent of the practice. Multiphonics were also widely used by John Coltrane, and jazz fl ...

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Multiphonic, Multiphonic - Technique, Multiphonic - How multiphonics work, Multiphonic - Notation, Multiphonic - Use in literature

Read more here: » Multiphonic: Encyclopedia II - Multiphonic - Use in literature

harmonic series: Encyclopedia II - Harmonic function - Remarks

The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over R: sums, differences and scalar multiples of harmonic functions are again harmonic. If f is a harmonic function on U, then all partial derivatives of f are also harmonic functions on U. In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are analytic, i.e. they can be locally expressed as power series. This is a general fact about elliptic operator ...

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Harmonic function, Harmonic function - Examples, Harmonic function - Remarks, Harmonic function - Connections with complex function theory, Harmonic function - Properties of harmonic functions, Harmonic function - The maximum principle, Harmonic function - The mean value property, Harmonic function - Liouville's theorem, Harmonic function - General theory

Read more here: » Harmonic function: Encyclopedia II - Harmonic function - Remarks

harmonic series: Encyclopedia II - Space elevator - Physics and structure

There are a variety of tether designs. Almost every design includes a base station, a cable, climbers, and a counterweight. Space elevator - Base station. The base station designs typically fall into two categories—mobile and stationary. Mobile stations are typically large oceangoing vessels, though airborne stations have been proposed as well. Stationary platforms are generally located in ...

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Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - Physics and structure

harmonic series: Encyclopedia II - Space elevator - Construction

The construction of a space elevator would be a vast project, requiring advances in engineering and physical technology. NASA has identified "Five Key Technologies for Future Space Elevator Development": Material for cable (e.g. carbon nanotube and nanotechnology) and tower Tether deployment and control Tall tower construction Electromagnetic propulsion (e.g. magnetic levitation) Space infrastructure and the development of space industry and economy Two different ways ...

See also:

Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - Construction

harmonic series: Encyclopedia II - Space elevator - Failure modes and safety issues

As with any structure, there are a number of ways in which things could go wrong. A space elevator would present a considerable navigational hazard, both to aircraft and spacecraft. Aircraft could be dealt with by means of simple air-traffic control restrictions, but impacts by space objects (in particular, by meteoroids and micrometeorites) pose a more difficult problem. Space elevator - Satellites. If nothing were done, essentially all satellites with perigees below the top of the elevator will eventuall ...

See also:

Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - Failure modes and safety issues

harmonic series: Encyclopedia II - Space elevator - Economics

Main article: space elevator economics With a space elevator, materials could be sent into orbit at a fraction of the current cost. Modern rocketry gives prices that are on the order of thousands of U.S. dollars per kilogram for transfer to low earth orbit, and roughly 20 thousand dollars per kilogram for transfer to geosynchronous orbit. For a space elevator, the price could be on the order of a few hundreds of ...

See also:

Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - Economics

harmonic series: Encyclopedia II - Space elevator - Political issues

One potential problem with a space elevator would be the issue of ownership and control. Such an elevator would require significant investment (estimates start at about US$5 billion for a very primitive tether), and it could take at least a decade to recoup such expenses. At present, only governments are able to spend in the space industry at that magnitude. Assuming a multi-national governmental effort was able to produce a working space elevator, many delicate political issues would remain to be solved. Which countries would ...

See also:

Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - Political issues

harmonic series: Encyclopedia II - Picardy third - Scientific explanation

It may be possible for the aural effect of the picardy third to be described scientifically - a major triad can be found in the 4th, 5th, and 6th harmonics of a major chord, while the minor triad can be found relatively higher in the 5th, 6th, and 7th harmonics of its respective chord. (In the language of the harmonic series, this is equivalent to saying the major chord occupies a lower 'location' in the harmonic series relative to the minor chord.) However, psychoacousticians cannot explain why this difference ...

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Picardy third, Picardy third - Example, Picardy third - History, Picardy third - Scientific explanation, Picardy third - Use in church music, Picardy third - Interpretation

Read more here: » Picardy third: Encyclopedia II - Picardy third - Scientific explanation

harmonic series: Encyclopedia II - Space elevator - History

The concept of the space elevator first appeared in 1895 when a Russian scientist Konstantin Tsiolkovsky was inspired by the Eiffel Tower in Paris to consider a tower that reached all the way into space. He imagined placing a "celestial castle" at the end of a spindle-shaped cable, with the "castle" orbiting Earth in a geosynchronous orbit (i.e. the castle would remain over the same spot on Earth's surface). The tower would be built from the ground up to an altitude of 35,790 kilometers above mean sea level (geostationary orbit). Comments from Nikola Tesla suggest that he may have also conceived such a tower. Tsiolkovsky's notes were sen ...

See also:

Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

Read more here: » Space elevator: Encyclopedia II - Space elevator - History

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