SHANTILAL SHAH GOVERNMENT ENGINEERING

COLLEGEPostal Address : New Sidsar Campus Post : Vartej, Sidsar, Bhavnagar-364060, Gujarat, IndiaContact : 0278-2445509,2445767 Fax : 0278-2445509Website: http://www.ssgec.ac.in/

Sr.no Roll no. Enrollment no.

Name

1. 3017 140430117019 JITIN J PILLAI2. 3010 140430117012 DODIYA PARTH3. 3007 140430117008 CHAUHAN ASHISH4. 3018 140430117020 JOSHI MEET5. 3002 140430117002 BAVISHI BIRJU

SEMESTER: 3RD BRANCH : INSTRUMENTATION & CONTROLBATCH : B1Group members :

Enrollment no.

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INTRODUCTION TO SOME SPECIAL FUNCTIONS

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Beta & Gamma functions Bessel function Error function & Complementary error function Heaviside's Unit Step Function Pulse Unit Height & Duration Sinusoidal pulse Rectangle function Gate function Dirac Delta function Signum function Saw tooth wave function Triangular wave function Half-wave Rectifed Sinusoidal function Full-wave Rectifed Sinusoidal function Square wave function

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Beta function

Bessel function

Gamma functions

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Error function

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Complementary error function

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Heaviside's Unit Step Function

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Pulse Unit Height & Duration

Sinusoidal Pulse

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Rectangle function

Gate function

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Dirac Delta function

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Signum function

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Saw tooth wave function

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Triangular wave function

Half-wave Rectifed Sinusoidal function

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Full-wave Rectifed Sinusoidal function

Square wave function

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FOURIER SERIES & FOURIER INTEGRAL

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Periodic function

Trigonometric series

Fourier series

Even and odd functions

Half-range Expansion

Applications of Fourier series: Forced

oscillations

Fourier integral

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A function f is said to be periodic with period P (P being a nonzero constant) if we have

for all values of x in the domain.

Periodic function

Trigonometric seriesA trigonometric series is a series of the form:

It is called a Fourier series if the terms and have the form:

where f is an integrable function.

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A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines.

Fourier series

With a Fourier series we are going to try to write a series representation for on in the form,

So, a Fourier series is, in some way a combination of the Fourier sine and Fourier cosine series.

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Even and Odd FunctionsThey are special types of functions

Even Functions

A function is "even" when:

f(x) = f(−x) for all x

Odd Functions

A function is "odd" when:

−f(x) = f(−x) for all x

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If a function is defined over half the range, say 0 to L, instead of the full range from −L to L, it may be expanded in a series of sine terms only or of cosine terms only. The series produced is then called a Half Range Fourier series.

Half-range Expansion

Conversely, the Fourier Series of an even or odd function can be analysed using the half range definition.

An even function can be expanded using half its range from

0 to L or−L to 0 or

L to 2L

That is, the range of integration is L.

Even function Half-range Expansion

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The Fourier series of the half range even function is given by:

for n=1,2,3,... , where

and bn=0

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Odd function Half-range Expansion

An odd function can be expanded using half its range from 0 to L, i.e. the range of integration has value L. The Fourier series of the odd function is:

Since ao = 0 and an = 0, we have:

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Applications of Fourier series : Forced Oscillations

Consider a mass-spring system as before, where we have a mass m on a spring with spring constant k , with damping c , and a force F (t) applied to the mass. Suppose the forcing function F (t) is 2L -- periodic for some L > 0 . The equation that governs this particular setup is :

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The general solution consists of the complementary solution xc , which solves the associated homogeneous equation mx ′′ + cx′ + kx = 0 , and a particular solution we call xp .

For c > 0 , the complementary solution xc will decay as time goes by. Therefore, we are mostly interested in a particular solution xp that does not decay and is periodic with the same period as F (t) .

We call this particular solution the steady periodic solution and we write it as xsp as before. What will be new in this section is that we consider an arbitrary forcing function F (t) instead of a simple cosine. For simplicity, let us suppose that c = 0 . The problem with c > 0 is very similar. The equation

has the general solution

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In the spirit of the last section and the idea of undetermined coefficients we first write

Then we write a proposed steady periodic solution x as

where an and bn are unknowns. We plug x into the differential equation and solve for an and bn in terms of cn and dn .

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Fourier integral

Let f(x) be a decaying (non-periodic ) function

• f(x) is given by “linear combination” of cos(λx) & sin(λx) with a continuous interval of frequencies:

all real numbers λ ≥ 0.

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Fourier integral:

Fourier integral formula:

Convergence Theorem:

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ReferencesINTERNET :

https://en.wikipedia.org/wiki/List_of_mathematical_serieshttps://en.wikipedia.org/wiki/Fourier_serieshttps://en.wikipedia.org/wiki/Fourier_integral_operatorpeople.math.gatech.edu/~xchen/teach/real.../Fourier_integral_intro.pdfmath.mit.edu/cse/websections/cse41.pdfwww.jirka.org/diffyqs/htmlver/diffyqsse29.htmlhttps://people.math.osu.edu/kwa.1/notes/512_2.7.pdfhttp://physics.bu.edu/~pankajm/PY501/fourier.pdfhttp://mathworld.wolfram.com/PeriodicFunction.htmlhttp://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspxhttp://www.fourier-series.com/

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BOOKS :

• Elementary Differential Equations (8th Edition), by W. E. Boyce and R. DiPrima, John Wiley (2005).

• Advanced Engineering Mathematics (9th Edition), by E. Kreyszig, Wiley-India (2013).

• Advanced Engineering Mathematics, by Ravish R. Singh and Mukul Bhatt, Mcgraw Hill Education (India) Pvt. Ltd.

References

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