Math and Arithmetic

Algebra

Geometry

Numerical Series Expansion

Probably the movement on a swing can be approximated by assuming that the magnitude of each swing will be a certain percentage of the previous swing (because of lost energy).

331332333

Numerical Series Expansion

Rapidity of convergence is a relative concept whose meaning comes from the "comparison test" for convergence. One version of the comparison test says that if (an) is convergent and abs(bn)<K*abs(an) for all n, for some K>0, then (bn) must be convergent too.

99100101

Probability

Numerical Series Expansion

Irrational Numbers

no. 3 = big water or sea

192021

Proofs

Numerical Series Expansion

Complex Numbers

Euler's formula states that eix = cos(x) + isin(x) where i is the imaginary number and x is any real number.

First, we get the power series of eix using the formula:

ez = Î£âˆžn=o zn/n! where z = ix. That gives us:

1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + (ix)8/8! + ...

which from the properties of i equals:

1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - x6/6! - ix7/7! + x8/8! + ...

which equals:

(1 - x2/2! + x4/4! - x6/6! + x8/8! - ...) + i(x - x3/3! + x5/5! - x7/7! + ...).

These two expressions are equivalent to the Taylor series of cos(x) and sin(x). So, plugging those functions into the expression gives cos(x) + isin(x).

Q.E.D.

Of course, were you to make x = Ð¿ in Euler's formula, you'd get Euler's identity:

eiÏ€ = -1

It depends on your definitionThe question presumes that someone has already given a definition of the exponential function for complex numbers. But has this definition been given? If so, what is this official definition?The answer above assumes that the exponential function is defined, for all complex numbers, by its power series. (Or, at least, that someone else has already proved that the power series definition is equivalent to whatever we're taking to be the official definition).

So the answer depends critically on what the definition of the exponential function for complex numbers is.

Suppose you know everything about the real numbers, and you're trying to build up a theory of complex numbers. Most people probably view it this way: Mathematical objects such as complex numbers are out there somewhere, and we have to find them and work out their properties. But mathematicians look at it slightly differently. If we can construct a thing which has all the properties we feel the field of complex numbers should have, then for all practical purposes the field of complex numbers exists. If we can define a function on that field which has all the properties we think exponentiation on the complex numbers should have, then that's as good as proving that complex numbers have exponentials. Mathematicians are comfortable with weird things like complex numbers, not because they have proved that they exist as such, but because they have proved that their existence is consistent with everything else. (Unless everything else is inconsistent, which would be a real pain but is very unlikely.)

So how do we construct this exponential function? One approach would be simply to define it by exp(x+iy) = ex(cos(y)+i.sin(y)). Then the answer to this question would be trivial: exp(iy) = exp(0+iy) = e0(cos(y)+i.sin(y)) = cos(y)+.sin(y). But you'd still have some work to do to prove things like exp(z+w) = exp(z).exp(w). Alternatively, you could define the exponential function by its power series. (There's a theorem that lets you calculate the radius of convergence, and that tells us the radius is infinite, i.e. the power series works everywhere.) Or maybe you could try something like proving that the equation dw/dz = w has a unique solution up to a multiplicative constant, and defining the exponential function to be the solution which satisfies w=1 at z=0.

Let z = cos(x) + i*sin(x)

dz/dx = -sin(x) + i*cos(x)

= i*(i*sin(x) + cos(x))

= i*(cos(x) + i*sin(x))

= i*(z)

therefore dz/dx = iz

(1/z)dz = i dx

INT((1/z)dz = INT(i dx)

ln(z) = ix+c

z = e(ix+c)

Substituting x=0, we get cos(0) + i*sin(0) = e(0i+c)

1+0 = e(0+c)

e(0+c) = 1

therefore c=0

z = eix

eix = cos(x) + i*sin(x)

Substituting x=pi, we get e(i*pi) = -1 + 0

e(i*pi) = -1

e(i*pi) + 1 = 0

qED

697071

Math and Arithmetic

Electronics Engineering

Mathematical Analysis

Numerical Series Expansion

The Fourier series is an expression of a pattern (such as an electrical waveform or signal) in terms of a group of sine or cosine waves of different frequencies and amplitude. This is the frequency domain.

The Fourier transform is the process or function used to convert from time domain (example: voltage samples over time, as you see on an oscilloscope) to the frequency domain, which you see on a graphic equalizer or spectrum analyzer.

The inverse Fourier transform converts the frequency domain results back to time domain. The use of transforms is not limited to voltages.

656667

Math and Arithmetic

Roman Numerals

Numerical Series Expansion

The 15, easy

636465

Consumer Electronics

Physics

Numerical Series Expansion

Similarities Between

Relationship Between Frequency and HarmonicsThe frequency is the fundamental frequency or the operating frequency, and the harmonics are multiples of that frequency which are generally of less amplitude. It's something that's inherent in non-sinusoidal oscillators, like sawtooth oscillators.

In digital, there's also usually a sinusoidal oscillator which is either built in or external, but the oscillations are converted into square waves to create a clock pulse which negates the harmonics.

There is a little problem in counting the harmonics and the overtones. So you find different statements in the internet. Some are wrong and some are right.

Scroll down to related links and look at "Calculations of Harmonics from Fundamental Frequency".

In digital, there's also usually a sinusoidal oscillator which is either built in or external, but the oscillations are converted into square waves to create a clock pulse which negates the harmonics.

There is a little problem in counting the harmonics and the overtones. So you find different statements in the internet. Some are wrong and some are right.

Scroll down to related links and look at "Calculations of Harmonics from Fundamental Frequency".

575859

Math and Arithmetic

Statistics

Numerical Series Expansion

- Each outcome must be classified as a success (p) or a failure (r),
- The probability distribution is discrete.
- Each trial is independent and therefore the probability of success and the probability of failure is the same for each trial.

282930

Statistics

Numerical Series Expansion

First i will explain the binomial expansion

495051

World Series

Warriors Book Series

Plural Nouns

Numerical Series Expansion

There was a strike that ended the baseball season in mid-August in 1994.

484950

Numerical Series Expansion

now i MAY be wrong, but ill go ahead:

Million

Billion

Trillion

Quadrillion

Pentillin

Sectillion

(Don't know the one for 7)

Octillion

And that's all I can figure, hope I was help XP

Edit: It goes by the common scientific naming of multiplicity, just like what is described above, but you don't use the prefixes used for geometry like what is described. You instead use the common prefixes used in naming chemical compounds. Here's the list:

Million

Billion

Trillion

Quadrillion

Quintillion

Sextillion

Septillion

Octillion

Nonillion

Decillion

Undecillion

Duodecillion

and so on and so forth...duodecillion is actually 10,000,000,000,000,000,000,000,000,000,000,000,000,000 so if you need to really count anything more than that you're probably at a molecular level and would simply call it 10 to the 39th power.

:)

474849

C Programming

Numerical Series Expansion

the Taylor series of sinx

394041

Statistics

Probability

Numerical Series Expansion

with n=15 as fixed, as p=0.5 changes to p=.05 the binmomial distribution will shift to the right.

394041

School Subjects

Florida State University

Essays

Numerical Series Expansion

The word divergence means separation, parting or moving in different directions.

Examples:

As we have grown older, my brother's and my taste in music has diverged - I prefer classic 60s rock whereas he likes more modern music.

When a laser beam moving in a straight is deflected in some angle, the laser is diverged.

394041

World Series

Mathematicians

Numerical Series Expansion

It follows the rules of the home team.

293031

World Series

Warriors Book Series

Plural Nouns

Numerical Series Expansion

Owing to business rivalry between the two leagues, especially in New York, and to personal animosity between Giants manager John McGraw and American League President Ban Johnson, the Giants declined to meet the champions of the "junior" or "minor" league. McGraw even went so far to say that his Giants were already the World Champions since they were the champions of the "only real major league".

232425

English Spelling and Pronunciation

Numerical Series Expansion

fourth

151617

Mathematical Analysis

Numerical Series Expansion

Yes. If the Maclaurin expansion of a function locally converges to the function, then you know the function is smooth. In addition, if the residual of the Maclaurin expansion converges to 0, the function is analytic.

293031

Roman Numerals

Numerical Series Expansion

34.

212223

C Programming

Numerical Series Expansion

PROGRAM :-

/* Runge Kutta for a set of first order differential equations */

#include

#include

#define N 2 /* number of first order equations */

#define dist 0.1 /* stepsize in t*/

#define MAX 30.0 /* max for t */

FILE *output; /* internal filename */

void runge4(double x, double y[], double step); /* Runge-Kutta function */

double f(double x, double y[], int i); /* function for derivatives */

void main()

{

double t, y[N];

int j;

output=fopen("osc.dat", "w"); /* external filename */

y[0]=1.0; /* initial position */

y[1]=0.0; /* initial velocity */

fprintf(output, "0\t%f\n", y[0]);

for (j=1; j*dist<=MAX ;j++) /* time loop */

{

t=j*dist;

runge4(t, y, dist);

fprintf(output, "%f\t%f\n", t, y[0]);

}

fclose(output);

}

void runge4(double x, double y[], double step)

{

double h=step/2.0, /* the midpoint */

t1[N], t2[N], t3[N], /* temporary storage arrays */

k1[N], k2[N], k3[N],k4[N]; /* for Runge-Kutta */

int i;

for (i=0;i

{

t1[i]=y[i]+0.5*(k1[i]=step*f(x,y,i));

}

for (i=0;i

{

t2[i]=y[i]+0.5*(k2[i]=step*f(x+h, t1, i));

}

for (i=0;i

{

t3[i]=y[i]+ (k3[i]=step*f(x+h, t2, i));

}

for (i=0;i

{

k4[i]= step*f(x+step, t3, i);

}

for (i=0;i

{

y[i]+=(k1[i]+2*k2[i]+2*k3[i]+k4[i])/6.0;

}

}

double f(double x, double y[], int i)

{

if (i==0)

x=y[1]; /* derivative of first equation */

if (i==1)

x= -0.2*y[1]-y[0]; /* derivative of second equation */

return x;

}

252627

Electronics Engineering

Electrical Engineering

The Difference Between

Numerical Series Expansion

Harmonics are multiples (thirds, fifths, etc) or divisions of frequencies. In radio, harmonics can be used carry additional signals on a single base frequency. It is the harmonics of an audio frequency that make a musical instrument unique. By damping a string at a half or third/fifth of its length a harmonic is easily heard. The truth is they were always there, the damping of the incident wave just brings them to the fore. Scroll down to related links and look at "Calculations of Harmonics from Fundamental Frequency".

272829

Numerical Series Expansion

why we use tailor series for calculating of error for a test with multiple variables justify it

192021

Statistics

Numerical Series Expansion

When you add together the probabilities of all outcomes in which the number of successes is equal to or fewer than the given number.

123

Math and Arithmetic

Statistics

Numerical Series Expansion

The binomial distribution is a discrete probability distribution. The number of possible outcomes depends on the number of possible successes in a given trial. For the Poisson distribution there are Infinitely many.

212223

Chennai

Numerical Series Expansion

Lower Kinder Garten & Upper Kinder Garten

91011

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