Team Quanta gladly presents all short questions of Methods of Mathematical Physics – II’s Chapter#4: Integral Transformation.
Q.1 what is meant by integral transformation?
Answer: A pair of functions related by an expressions of the form;
$$g\left(\alpha\right)=\int_{a}^{b}f\left(t\right)k\left(\alpha,t\right)dt$$
The function $g\left(\alpha\right)$ is called integral transformation of f(t) by the kernel . The operation may be mapping of a function f(t) in t space into another function $k\left(\alpha,t\right)$ in space.
Q.2 what are the physical significance of integral transformation?
Answer: It gives the relationship between two spaces like real space and momentum space. It gives the relationship between two quantities like time and frequency.
Q.3 Write down the applications of Integral Transform?
Answer: It has the following applications;
- Very important in mathematics and physics.
- Tools of solving ordinary differential equation, partial differential equation and integral equations.
- They helps in handling and applying special functions
- Laplace in very important
- Fourier transformation is very used in music, for infinite set of frequencies
Q.4 Define Fourier transforms?
Answer: The;
$$F\left(w\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\ F\left(t\right)e^{iwt}dt$$
This equation is called Fourier transforms.
Q.4 what is Fourier cosine transform?
Answer: The;
$$F_c(\omega )=\sqrt{\frac{2}{\pi }} \int_{+\infty }^{0}F_c(t)cos\omega t dt$$
This is called Fourier cosine transform.
Q.6 what is meant by Fourier sine transform?
Answer: The;
$$F_s(\omega )==\sqrt{\frac{2}{\pi}}\int_{+\infty}^{0}f(t)dt \ sin \omega \ tdt$$
This equation is called Fourier sine transform.
Q.7 what do you meant by inverse Fourier transform?
Answer: The;
$$F\left(t\right)=\frac{1}{\sqrt{2\pi}}= \int_{+\infty }^{-\infty} F_c(t)cos\omega t dt$$
Q.8 Prove inverse cosine Fourier transform?
Answer: Proof;
$$F_c(\omega )=\sqrt{\frac{2}{\pi }} \int_{+\infty }^{0}F_c(t)cos\omega t dt$$
$$F_c(\omega )=\sqrt{\frac{2}{\pi }} \int_{+\infty }^{0}F_c(x)cos\omega x dx$$
$$F_c\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{+\infty} F_c(\omega )e^{iwx}dx$$
is even (odd will be vanish)
This is called inverse cosine Fourier transform.
Q.9 Write the equation of inverse sine Fourier transforms?
Answer: Similarly, the sine inverse transform will be,
$$F_s\left(x\right)=\sqrt{\frac{2}{\pi}}\int_{0}^{+\infty }F_s(\omega )sin\omega xdx$$
This is called inverse sine Fourier transform.
Q.10 Find the Fourier transform of F(t)=t, -1 to +1 ?
Answer: Solution;
$$F\left(\omega \right)=\sqrt{\frac{2}{\pi}}\int_{-1}^{+1 } 1e^{iwt}dt$$
$$F\left(\omega \right)=\sqrt{\frac{2}{\pi}}\int_{-1}^{+1 } e^{iwt}dt$$
Q.12 what are the properties of Dirac Delta function?
Answer: They are following properties such as;
- $\delta\left(-x\right)=\delta x$
- $\delta^\prime\left(x\right)=\delta^\prime(-x)$
- $x\delta\left(x\right)=0$
- $x\delta’\left(x\right)=-\delta\left(x\right)$
- $\delta\left(ax\right)=\frac{\delta\left(x\right)}{\left|a\right|}$
- $\delta\left(x^2-a^2\right)=\frac{\delta\left(x-a\right)+\delta\left(x+a\right)}{\left|2a\right|}$
- $\int_{-\infty}^{+\infty}{\delta\left(-a-x\right)\delta\left(x-b\right)dx=\delta\left(a-b\right)}$
- $F\left(x\right)\delta\left(x-a\right)=F\left(a\right)\delta\left(x-a\right)$
Q.13 what is delta function used for?
Answer: The Dirac Delta Function is an important mathematical object that simplifies calculations required for the studies of electron motion and propagation. It is not a really a function but a symbol for physicists and engineers to represent some calculations.
Q.14 Is Delta an energy symbol?
Answer: The is an even signal. It is an example of neither energy nor power (NENP) signal.
Q.15 what do you mean by Fourier Integral?
Answer: A formula for the decomposition of a non-periodic function into harmonic components whose frequencies range over a continuous set of values. If a function F(x) satisfies the Dirichlet condition on every finite interval and if the integral converges then the formula was first introduced in 1811.
Q.16 why Fourier integral is used?
Answer: The straightforward application of the Fourier integral to determine the response of a linear invariable circuit to an arbitrary impressed force is reviewed.
When a Fourier integral representation of the impressed force exists and the system starts from rest the problem is routine.
Q.17 what is difference between Fourier integral and Fourier transform?
Answer: Fourier transform of a function F is the function Ff is defined by;
$$Ff\left(w \right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{f\left(t\right)e^{-iwt}}dt$$
Fourier integral is any integral of the form $\int_{-\infty}^{+\infty}{y\left(\omega \right)e^{iwt}dw}$
Fourier integral of a function is any Fourier integral that satisfies;
$$x\left(t\right)=\int_{-\infty}^{+\infty}{y\left(\omega\right)e^{iwt}dw}$$
Q.18 what is use of Fourier transform?
Answer: The Fourier transform can be used to interpolate functions and to smooth signals. For example, in processing of pixelated images, the high spatial frequency edges of pixels can easily be removed with the aid of a two-dimensional Fourier transform.
Q.19 what is Fourier integral transform?
Answer: The Fourier transform uses an integral or continuous sum that exploits properties of sine and cosine to recover the amplitude and phase of each sinusoidal in a Fourier series. The inverse Fourier transform recombines these waves using a similar integral to reproduce the original function.
Q.20 why is Fourier transform complex?
Answer: The complex versions have a complex time domain signal and a complex frequency domain signal. The real versions have a real time domain signal and two real frequency domain signals. Both positive and negative frequencies are used in the complex cases while only positive frequencies are used for the real time.
Q.21 Write the exponential form of Fourier integral?
Answer: The exponential form of Fourier integral is;
$$F\left(x\right)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}{F\left(t\right)e^{-iwt}dw\int_{-\infty}^{+\infty}{F\left(t\right)e^{iwt}dt}}$$
Q.22 what are the formulas of Fourier transform of derivative?
Answer: The;
$$F\left(\omega\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(t\right)e^{+iwt}}dt\$$
And Fourier transform of $F’\left(t\right)=\frac{dF}{dt}$ is;
$$F_{1\left(w\right)}=\frac{1}{\sqrt{2\pi}}\left[F\left(t\right)e^{iwt}|\begin{matrix}+\infty\-\infty\\end{matrix}-\int_{-\infty}^{+\infty}{F\left(t\right)e^{iwt}\left(iw\right)dt}\right]$$
$$F_{1\left(w\right)}=\frac{1}{\sqrt{2\pi}}\left[0-\int_{-\infty}^{+\infty}{F\left(t\right)e^{iwt}\left(iw\right)dt}\right]$$
$$F_{1\left(w\right)}=\left[-\frac{iw}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(t\right)e^{iwt}dt}\right]$$
$$F_{1\left(w\right)}=\left(-iw\right)F\left(w\right)$$
$$F_2\left(w\right)=(-i)^2F(w)$$
$$F_3\left(w\right)=(-i)^3F(w)$$
$$\vdots$$
$$\vdots$$
$$F_n\left(w\right)=(-i)^nF(w)$$
For nth derivative.
Q.23 Prove; $F_1\left(k\right)=\left(-ik\right)F\left(k\right)\ ?$
Answer: Proof;
$$F\left(k\right)=F\left(f\left(t\right)\right)$$
$$F_1\left(k\right)=F\left(f\left(t\right)\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(t\right)e^{ikt}dt}$$
$$F_(\frac{\partial F}{\partial t})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{\frac{\partial}{\partial t}F\left(t\right)e^{ikt}dt}$$
$$F_1\left(k\right)=\frac{1}{\sqrt{2\pi}}\left[e^{ikt}F\left(t\right)\middle|\begin{matrix}+\infty\-\infty\\end{matrix}-\int_{-\infty}^{+\infty}{F\left(t\right)e^{ikt}dt}\right]$$
$$F_1\left(k\right)=F\left(f\left(t\right)\right)=0+\frac{-ik}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(t\right)e^{ikt}dt}$$
$$F_1\left(k\right)=\left(-ik\right)F\left(k\right)$$
Q. 24 what are the damped harmonic oscillation in Fourier transform?
Answer: Consider DHO acted upon an external force g(t), it is given by differential equation;
$$x\left(t\right)+2\alpha x\left(t\right)+w_0^2x\left(t\right)=F\left(t\right)$$
Where $F\left(t\right)=\left(\frac{1}{m}\right)g\left(t\right)$ Fourier transform of F(t).
$$F\left(w\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(t\right)e^{iwt}dt}$$
$$F\left(t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{F\left(w\right)e^{-iwt}dw}$$
Now taking Fourier transform of each term of differential equation;
$$x\left(t\right)=\frac{F\left(t\right)}{\left(w_0^2-w^2\right)-2\alpha iw}$$
Q.27 what is the Laplace transform of coshkt?
Answer: The Laplace transform of coshkt is;
$$\mathcal{L}\left{\cos{hkt}\right}=\frac{s}{\left(s^2-k^2\right)}$$
Q.28 what is the Laplace transform of ekt
Answer: The Laplace transform of is;
$$\mathcal{L}\left{e^{kt}\right}=\frac{1}{\left(s-k\right)}$$
Q.29 Write the Laplace transform of sinkt?
Answer: The Laplace transform of sinkt is;
$$\mathcal{L}\left{\sin{kt}\right}=\frac{ik}{\left(s^2+k^2\right)}$$
Q.30 what are the applications of Laplace transform?
Answer: The main applications of Laplace transforms is in converting, differential equations into simpler forms that may solved more easily. It will be seen for instance, that coupled differential equations with constant coefficient transform to simultaneous linear algebraic.
Q.31 what are the first derivative of Laplace transform?
Answer: The 1st derivative is;
$$\mathcal{L}\left{F’\left(t\right)\right}=-F\left(0\right)+s\mathcal{L}F\left(t\right)$$
Q.32 what are the 2nd derivative of Laplace transform?
Answer: The 2nd derivative is;
$$Ⅎc{L}{F^{”}├(t┤)=s^2Ⅎc{L}├{F├(t┤)┤}-sF├(0┤)-F^'(0)$$
Q.34 what are the following properties of Laplace transform?
Answer: The following properties of Laplace transform are given as;
- Substitution
- Translation property.
Q.35 what is the substitution property of Laplace transform?
Answer: If we replace s by s-a then the definition of Laplace transform becomes;
$$F\left(s-a\right)=\mathcal{L}\left[e^{at}F\left(t\right)\right]$$
So, the replacement of s by sa corresponding to multiply by F(t) by
$$\mathcal{L}\left[e^{at}\sin{kt}\right]=\frac{k}{\left(s-a\right)^2+k^2}$$
$$\mathcal{L}\left[e^{at}\cos{kt}\right]=\frac{s-a}{\left(s-a\right)^2+k^2}$$
Q.37 what do you mean by wave equation?
Answer: The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields. They occur in classical physics such as mechanical waves (water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).
Q.38 how do you derive a wave equation?
Answer: The wave equation is derived by applying F=ma to an infinitesimal length dx of string. We picture our little length of string as object up and down in simple harmonic equation motion, which we can verify by finding the net force on it as follows.
Q.39 what do you meant by Heaviside Shifting Theorem?
Answer: The second shifting theorem is a useful tool when faced with the challenge of taking the Laplace transform of the product of a shifted unit step function (Heaviside function) with another shifted function. The Laplace transform is very useful in finding ordinary differential equations.
Q.40 Is Heaviside function odd?
Answer: H(x) is not an odd function. Addendum in terms of the Iverson bracket, [P]=1. If P is true 0, otherwise, the Heaviside step function is $\ H\left(x\right)=\left[x>0\right]+12\left[x=0\right]$ these are different conventions for H(0).
Q.41 what is the difference between Heaviside step function and unit step function?
Answer: The Heaviside step function or the unit step function, usually denoted by H or (but sometimes u,1 or 1), is a step function, named after Olivev Heaviside (1850-1925), the value of which is zero for negative arguments and one for positive arguments.
Q.42 what is unit step function?
Answer: The definition of continuous time unit step function, it is clear that the unit step function is zero when the time(t) is less than zero and when the time(t) is greater than or equal to zero, then u(t) is unity.
Q.43 why is a Heaviside function also called a step function?
Answer: As it can be observed, a Heaviside function can only have values of 0 and 1. The function is always equal to zero before arriving to a certain value t=c at which it turns on and jumps directly into having a value of 1 and so, it jumps in a step size of one unit.
Q.44 Is unit step function finite?
Answer: A step function can take only a finite number of values.
Q.45 Is unit step function causal?
Answer: It is not a causal.
Q.46 what is impulse force Delta?
Answer: An impulse function often called Dirac Delta function, is defined not by its values but by its behaviour in a limit and by its behaviour under integration. The Dirac Delta function provides a model for a force that concentrates a large amount of energy over a short time interval.
Q.47 Define inverse Laplace transform?
Answer: Now we develop an expression for the inverse transform appearing in the equation;
$$\mathcal{L}\left{F\left(t\right)\right}=F\left(s\right)$$
$$\mathcal{L}^{-1}\left{F\left(s\right)\right}=F\left(t\right)$$
This equation is the equation of inverse Laplace transform.
Q.48 what is convolution theorem?
Answer: We shall empty convolution to solve differential equation to normalized momentum wave function and to investigate transfer function. Consider two functions F(t) and G(t) respectively, then relation;
$$F+G=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}G\left(y\right)F\left(x-y\right)dy$$
Is known as convolution of two functions F and G over the interval $\left(-\infty,+\infty\right)$
Q.49 what are the applications of convolution theorem?
Answer: It has following applications;
- It is used to normalize momentum
- To integrate transform function
- Probability theorem is to determine probability density of two random independent variables.
Q.50 what is the definition of Hermite polynomial?
Answer: The Hermite polynomial $H_n\left(x\right)$ may be defines by the generating function.
$$g\left(x,t\right)=e^{t^2+2tx}=\sum_{n=0}^{+\infty}{H_n\left(x\right)\frac{t^n}{n!}}$$
Q.51 what is Delta in impulse?
Answer: In mathematics, Dirac Delta distribution ( distribution) also known as the unit impulse, is a generalized function or distribution over the real numbers whose value is zero everywhere except at zero and whose integral over the entire real line is equal to one.
Q.52 what is the unit impulse response?
Answer: The unit impulse response of the system is simply the derivative;
$$y\partial\left(t\right)=dyy\left(t\right)dt$$
The unit step response is a zero state response. That is the limit initial conditions at t=0 are all zero. The unit impulse response is , therefore also a zero a state response.
Q.54 what are the alternate representation of Rodrigues?
Answer: The Rodrigues representation of is;
$$H_n\left(x\right)=\left(-1\right)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$
Q.55 Define Laguerr’s equation?
Answer: The differential equation;
$$X\frac{d^2y}{dx^2}+\left(1-x\right)\frac{dy}{dx}+ly=0\$$
Where l is constant and the equation is called Laguerr’s equation.
Q.56 what are the main application of Laguerr’s equation?
Answer: The most important application of Laguerr’s polynomial is in the solution of Schrodinger equation;
For the Hydrogen atom
$$-\frac{h^2}{2m}\nabla^{2\mathrm{\Psi}}-\frac{ze^2}{4\pi\varepsilon_0r}\mathrm{\Psi}=E\mathrm{\Psi}$$
In which z=1 for hydrogen atom.
Q.57 Write the formula of Laguerr’s polynomial?
Answer: For Laguerr’s polynomial;
$$L_l\left(x\right)=\sum_{n=0}^{+\infty}\frac{\left(-1\right)^nl!x^2}{\left(n!\right)^2\left(l-n\right)!}$$
