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Thursday, March 5, 2020

M4.6 Some identities on Analytic and harmonic functions

MathJax example 01. If u and v are harmonic functions show that (uyvx)+i(ux+vy) is analytic.
Ans:
Let P=(uyvx) and Q=(ux+vy).
Now we have to show that P+iQ is analytic, i.e., we have to show that P and Q will satisfies CR-equations
PxQy=x(uyvx)y(ux+vy)

PxQy=2uxy2vx22uyx2vy2=(2vx2+2vy2)=0

[because 2vxy=2vyx and since u and v are harmonic, u and v satisfies Laplace equations]. Hence PxQy=0Px=Qy
Similarly, consider
Py+Qx=y(uyvx)+x(ux+vy)

Py+Qx=2uy22vyx+2ux2+2vxy=(2ux2+2uy2)=0

[because 2uxy=2uyx and since u and v are harmonic, u and v satisfies Laplace equations]. Hence Qx+Qy=0Qx=Qy
Hence P+iQ is analytic.

02. If f(z) is analytic, show that (2x2+2y2)|f(z)|2=4|f(z)|2 is analytic.
Ans:
Let f(z)=u+iv be analytic. |f(z)|=u2+v2|f(z)|2=u2+v2=ϕ<\br> Hence, we need to show that ϕxx+ϕyy=4|f(z)|2.<\br> ϕ=u2+v2ϕx=2uux+2vvxϕxx=2[uuxx+u2x+vvxx+v2x](1) ϕ=u2+v2ϕy=2uuy+2vvyϕyy=2[uuyy+u2y+vvyy+v2y](2) Now add (1) and (2)
ϕxx+ϕyy=2[u(uxx+uyy)+v(vxx+vyy)+u2x+v2x+u2y+v2y](3) Since f(z) is analytic, u and v are harmonic uxx+uyy=0,vxx+vyy=0, also using CR-equations in (3), we get.
ϕxx+ϕyy=2[2u2x+2v2x]=4[u2x+v2x](4) But f(z)=ux+ivx|f(z)|=u2x+v2x, hence we can conclude that (2x2+2y2)|f(z)|2=4|f(z)|2
03. If f(z) is a regular analytic, show that (x|f(z)|)2+(y|f(z)|)2=|f(z)|2
Ans:
Let f(z)=u+iv be analytic. |f(z)|=u2+v2=ϕ|f(z)|2=u2+v2=ϕ2<\br> Consider ϕ2=u2+v2 2ϕϕx=2uux+2vvxϕϕx=uux+vvx(1) similarly2ϕϕy=2uuy+2vvyϕϕy=uuy+vvy(2) Squaring and adding both (1) and (2), we will get ϕ2(ϕ2x+ϕ2y)=(uux+vvx)2+(uuy+vvy)2 =(u2u2x+v2v2x+2uvuxvx)+(u2u2y+v2v2y+2uvuyvy) Since f(z) is analytic, u and v satisfies CR-equations. ϕ2(ϕ2x+ϕ2y)=u2(u2x+v2x)+v2(u2x+v2x)=(u2+v2)(u2x+v2x)=ϕ2(u2x+v2x) Hence, we get ϕ2x+ϕ2y=u2x+v2x Also we have f(z)=u+ivf(z)=ux+ivx|f(z)|=u2x+v2x, hence above equation will become ϕ2x+ϕ2y=|f(z)|2

Wednesday, February 19, 2020

M4.5 Application on Analytic Functions

MathJax example Let vector field V=vxˆi+vyˆj represents velocity vector field. If the fluid is irrotational then ×V=0. Also the velocity of an irrotational flow can be represented by gradient of a scalar function ϕ. i.e., V=ϕ=ϕxˆi+ϕyˆj(1). ϕ is called as velocity potential and the family of curves ϕ(x,y)=c are called as equipotenttial lines

If the fluid is incompressible then divV=0, using (1), we will get 2ϕ=0, hence ϕ is harmonic. In other words an incompressible, irrotational flow is governed by Laplace's equation.

If ϕ is harmonic then there exists conjugate harmonic function ψ(x,y) called as stream function such that w(z)=ϕ(x,y)+iψ(x,y) is analytic. Where the family of curves ψ(x,y)=c are called as stream lines
From orthogonal property we can say that stream lines and equipotential lines are orthogonal as shown in the following figure.

Problems
01. Find the complex potential of a fluid flow if the stream function is given by ψ=yx2+y2. Also find the velocity potential ϕ.
Ans:
Let complex potential of a fluid is given by f(z)=ϕ(x,y)+iψ(x,y)f(z)=ϕx+iψx
Since f(z) is analytic, ϕ,ψ satisfies CR-equationsf(z)=ψy+iψx
Given ψ=yx2+y2ψx=2xy(x2+y2)2&ψy=y2x2(x2+y2)2
f(z)=y2x2(x2+y2)2+i2xy(x2+y2)2
Now put x=z,y=0f(z)=z2(z2)2+i0f(z)=1z2f(z)=1z
Now put z=x+iyf(z)=1x+iy=xiyx2+y2=xx2+y2iyx2+y2
Hence ϕ=xx2+y2
Mathematica Notebook Sourcecode which shows the above problem Graphically.
02. If ϕ+iψ represents the complex potential of an electric field if ψ=(x2y2)+xx2+y2 find the complex potential as a function of z and also find ϕ.
Ans:
Let complex potential of a fluid is given by f(z)=ϕ(x,y)+iψ(x,y)f(z)=ϕx+iψx
Since f(z) is analytic, ϕ,ψ satisfies CR-equationsf(z)=ψy+iψx
Given ψ=(x2y2)+xx2+y2ψx=2x+y2x2(x2+y2)2&ψy=2y2xy(x2+y2)2
f(z)=2y2xy(x2+y2)2+i(2x+y2x2(x2+y2)2)
Now put x=z,y=0f(z)=i(2z1z2)f(z)=i(z2+1z)
Now put z=x+iyf(z)=i((x+iy)2+1x+iy)=i((x2y2+i2xy)+xiy(x+iy)(xiy))=(2xy+yx2+y2)+i((x2y2)+xx2+y2)
Hence ϕ=2xy+yx2+y2

Saturday, February 15, 2020

M2.1 Vector Calculus Introduction

MathJax example Scalars: The physical quantities having only magnitude are called as scalars. Ex: length, mass, time, speed, a=4,c=π
Vectors: The physical quantities having both magnitude and direction are called as vectors. Ex: velocity, force, v=aˆi+bˆj A vector in three dimension is of the form v=aˆi+bˆj+cˆk, where ˆi,ˆj,ˆk are the unit vectors in the direction of X,Y and Z respectively.
Addition of two vectors
Let v1=aˆi+bˆj,v2=cˆi+dˆj then v1+v2=(a+c)ˆi+(b+d)ˆj and v1v2=(ac)ˆi+(bd)ˆj
Following is the example for vector addition, created using geogebra.

Friday, February 14, 2020

M4.4 Problems on Analytic Functions

MathJax example
01. Show that w=z+ez is analytic and find dwdz.
Ans: w=f(z)=z+ezu+iv=(x+iy)+e(x+iy)=(x+iy)+exeiy=(x+iy)+ex(cosy+isiny) u=x+excosy;v=y+exsiny ux=1+excosy;vx=exsiny uy=exsiny;vy=1+excosy From the above set of equations we have ux=vy;vx=uy, hence u,v satisfies the CR-equations, Hence f is analytic
To find dwdz: Differentiate w=f(z)=u+iv with respect to x (derivative horizontally)
dwdz=f(z)=ux+ivx=1+excosy+iexsiny=1+ex(cosy+isiny)=1+exeiy=1+ex+iy=1+ez
02. Show that w=sinz is analytic and hence find dwdz.
Ans: w=f(z)=sinzu+iv=sin(x+iy)=sinxcosiy+cosxsiniy (we know that sinix=isinhx,cosix=coshx) u+iv=sinxcoshy+icosxsinhy u=sinxcoshy;v=cosxsinhy ux=cosxcoshy;vx=sinxsinhy uy=sinxsinhy;vy=cosxcoshy From the above set of equations we have ux=vy;vx=uy, hence u,v satisfies the CR-equations, Hence f is analytic
03. Construct the analytic function f(z)=u+iv given that u=log(x2+y2)
Ans:
Let f(z)=u+iv, since f is analytic, u,v satisfies CR-equations and we have u only, hence we have f(z)=ux+ivx=uxiuy Now u=log(x2+y2)=log(x2+y2)1/2=12log(x2+y2) ux=122xx2+y2uy=122yx2+y2 f(z)=122xx2+y2i122yx2+y2 Now substitute x=z and y=0 in the above equation we get f(z)=1/zf(z)=log(z) Verified using Maple

04. Construct the analytic function f(z)=u+iv given that u=e2x(xcos2yysin2y)
Ans:
Let f(z)=u+iv, since f is analytic, u,v satisfies CR-equations and we have u only, hence we have f(z)=ux+ivx=uxiuy ux=2e2x(xcos2yysin2y)+e2x(cos2y)uy=e2x(2xsin2ysin2y2ycos2y) f(z)=(2e2x(xcos2yysin2y)+e2x(cos2y))+i(e2x(2xsin2ysin2y2ycos2y)) Now substitute x=z and y=0 in the above equation we get f(z)=2e2zz+e2z+i(0)=e2z(1+2z)f(z)=(1+2z)e2z2e2z2+c=ze2z+c
05. Construct the analytic function f(z)=u+iv given that v=ex(xsiny+ycosy)
Ans:
Let f(z)=u+iv, since f is analytic, u,v satisfies CR-equations and we have v only, hence we have f(z)=ux+ivx=vy+ivx vx=ex(xsiny+ycosy)+ex(sin(y))vy=ex(xcosy+cosyysiny) f(z)=(ex(xcosy+cosyysiny))+i(ex(xsiny+ycosy)+ex(sin(y))) Now substitute x=z and y=0 in the above equation we get f(z)=ez(z+1)f(z)=(z+1)ezez+c=zez+c
06. Construct the analytic function f(z)=u+iv given that u=x4y42xx2+y2-(OQP,MQP)
Ans:
Let f(z)=u+iv, since f is analytic, u,v satisfies CR-equations and we have u only, hence we have f(z)=ux+ivx=uxiuy ux=(x2+y2)(4x32)(x4y42x)2x(x2+y2)2uy=(x2+y2)(4y3)(x4y42x)2y(x2+y2)2 f(z)=((x2+y2)(4x32)(x4y42x)2x(x2+y2)2)i((x2+y2)(4y3)(x4y42x)2y(x2+y2)2) Now substitute x=z and y=0 in the above equation we get f(z)=2z5+2z2z4i(0)=2z+1z2f(z)=z22z+c
07. Construct the analytic function f(z)=u+iv given that v=(rk2r)sinθ-(MQP)
Ans:
Let f(z)=u+iv, since f is analytic, u,v satisfies CR-equations and we have u only, hence we have f(z)=f(reiθ)eiθ=(ur+ivr)f(z)=eiθ(ur+ivr)=eiθ(vθr+ivr) vr=(1+k2r2)sinθvθ=(rk2r)cosθ f(z)=eiθ[1r(rk2r)cosθ+i(1+k2r2)sinθ] Now substitute r=z and θ=0 in the above equation we get f(z)=1k2zf(z)=z+k2z+c
08. Construct the analytic function f(z)=u+iv given that uv=ex(cosysiny)
Ans:
uv=ex(cosysiny), Differentiate with respect to x uxvx=ex(cosysiny)(1) Differentiate with respect to y uyvy=ex(sinycosy) Using CR-equations, we can write the same as vxux=ex(sinycosy)(2) (1)+(2)2vx=2exsiny,vx=exsiny (1)(2)2ux=2excosy,ux=excosy
Do it yourself
01. Construct the analytic function f(z)=u+iv given that uv=(xy)(x2+4xy+y2).
02. Construct the analytic function f(z)=u+iv given that u+v=1r2(cos2θsin2θ).
03. Construct the analytic function f(z)=u+iv given that sum of its real and imaginary parts is x3y3+3xy(xy).

Thursday, February 13, 2020

M4.3 Harmonioc Property & Orthogonal Property of a Complex Function

MathJax example

Harmonic Function

A function ϕ is said to be harmonic if it satisfies the Laplace equation 2ϕ=0. i.e., 2ϕx2+2ϕy2=0CartesianForm 2ϕr2+1rϕr+1r2ϕ2θ2=0PolarForm

Harmonic Property

The real and imaginary parts of an analytic function are harmonic
Proof: Cartesian Form
Let f(z)=u(x,y)+iv(x,y) be analytic. We have to show that u and v satisfies the Laplace equations. i.e., 2ux2+2uy2=0 & 2vx2+2vy2=0
Since f(z) is analytic, u and v should satisfies CR-equations ux=vx(1) vx=uy(2) Differentiate (1) with respect to x and differentiate (2) with respect to y partially, we get 2ux2=2vxy;2vyx=2uy2 But we know that 2vxy=2vyx Hence 2ux2=2uy22ux2+2uy2=0 Similarly, by differentiating (1) with respect to y and differentiate (2) with respect to x partially, we get 2uyx=2vy2;2vx2=2uxy Since 2vxy=2vyx, 2vy2=2vx22vx2+2vy2=0 Hence the real and imaginary parts of an analytic function are harmonic.
Polar Form
Let f(z)=u(r,θ)+iv(r,θ) be analytic. We have to show that u and v satisfies the Laplace equations. i.e., 2ur2+1rur+1r2u2θ2=0 and 2vr2+1rvr+1r2v2θ2=0. Since f(z) is analytic u and v should satisfies CR-equations ur=1rvθ vr=1ruθ Differentiate (1) with respect to r and differentiate (2) with respect to θ partially, we get r2ur2+ur=2vrθ;r2vθr=2uθ2 It is true that 2vrθ=2vθr, hence by substituting one in another we get r2ur2+ur=1r2uθ2 Divide throughout by r we get 2ur2+1rur+1r2u2θ2=0. Hence u is Harmonic. Similarly we can obtain 2vr2+1rvr+1r2v2θ2=0 by differentiating (1) with respect to θ and differentiate (2) with respect to r partially.

Orthogonal Property

If f(z)=u+iv is analytic then the family of curves u(x,y)=c1,v(x,y)=c2,c1 and c2 being constants, intersect each other orthogonally.
Proof: Cartesian Form
We say that two curves cuts orthogonally at some point if the product of the slope of the tangents (dydx) drawn to those curve at that point is equal to 1.
Let m1 be slope of the tangent drawn to the curve u(x,y)=c1. Differentiate u(x,y)=c1 with respect to x by treating y as function of x, we get ux+uydydx=0m1=dydx=ux/uy Similarly let m2 be slope of the tangent drawn to the curve v(x,y)=c2. Differentiate v(x,y)=c2 with respect to x by treating y as function of x, we get m2=dydx=vx/vy m1m2=ux/uyvx/vy Using CR-equation in the above equation we get m1m2=vy/uyuy/vy=1 Hence the curves intersect orthogonally.
  • Converse is not true, for example the curves u=x2y and v=x2+2y2 intersects orthogonally but it does not satisfies CR-equations, hence the function f(z)=u+iv is not analytic.

Wednesday, February 12, 2020

M4.2 Cauchy Riemannian Equations in Cartesian Form/Polar Form

MathJax example

Cauchy Riemannian Equations in Cartesian Form - (VTU 2020,2019,2018,2017)

The necessary condition for a complex function f(z)=u(x,y)+iv(x,y) is said to be analytic is that, there exists four first order continuous partial derivatives ux,uy,vx,vy such that it satisfies the conditions ux=vx vx=uy Proof: Let f(z)=u(x,y)+iv(x,y) be a complex function and is analytic at z=x+iy. According to definition of Analytic function we have f(z)=limδzzf(z+δz)f(z)δz. Let δz=δx+iδy then f(z+δz)=u(x+δx,y+δy)+iv(x+δx,y+δy) f(z)=[u(x+δx,y+δy)u(x,y)]+i[v(x+δx,y+δy)v(x,y)]δx+iδy f(z)=limδz0[u(x+δx,y+δy)u(x,y)]δx+iδy+ilimδz0[v(x+δx,y+δy)v(x,y)]δx+iδy Case 01: If z approaches horizontally, i.e., δy=0 then δz=δx f(z)=limδx0[u(x+δx,y)u(x,y)]δx+ilimδx0[v(x+δx,y)v(x,y)]δx f(z)=ux+ivx(1)
Case 02: If z approaches vertically, i.e., δx=0 then δz=iδy f(z)=limiδy0[u(x,y+δy)u(x,y)]iδy+limiδy0[v(x,y+δy)v(x,y)]δy f(z)=iuy+vy(2) Comparing (1) and (2) we will get ux=vx,vx=uy

Cauchy Riemannian Equations in Polar Form - (VTU 2020,2019,2018,2017)

The necessary condition for a complex function f(z)=u(r,θ)+iv(r,θ) is said to be analytic is that, there exists four first order continuous partial derivatives ur,uθ,vr,vθ such that it satisfies the conditions ur=1rvθ vr=1ruθ Proof:Let f(z)=u(r,θ)+iv(r,θ) be a complex function and is analytic at z=reiθ. f(z)=f(reiθ)=u(r,θ)+iv(r,θ)=u+iv Since f is an Analytic function, f(z) is exists and unique. Differentiate f with respect to r f(z)=f(reiθ)eiθ=ur+ivr(1) Differentiate f with respect to θ f(z)=f(reiθ)reiθi=uθ+ivθ(2) Put (1) in (2), we will get ri(ur+ivr)=uθ+ivθ irurrvr=uθ+ivθ Comparing real and imaginary part of the above equation we get ur=1rvθ vr=1ruθ

M4.1 Complex Analysis-Introdution

MathJax example

Review of basic definitions & Key points


  • The numbers are of the form x+iy are called as complex numbers, where x & y are Real numbers and i=1.
  • i=1 is obtained from the equation x2=1.
  • 1,i,i2,i3 are the fourth roots of the unity and i2=1,i3=i,i4=1.
  • Geometrically a complex number x+iy is a point in Cartesian plane as shown in the following figure
  • From the figure we can say that r=\sqrt{x^2+y^2} and \theta=tan^{-1}{y/x} are called as modulus and amplitude of the complex number z denoted by |z| and amp(z) or arg(z) respectively
  • z\bar=x-iy is the complex conjugate of the complex number z=x+iy

Euler's Formula

We know the following series 
  • e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...
  • \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-+...
  • \sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-+...
If we multiply i to the above equation we get
  • i \sin(x)=ix-\frac{ix^3}{3!}+\frac{ix^5}{5!}-+... Now
  • \cos(x)+i \sin(x)=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-+... \,\,\,\,\,\,\,\,\,\, (1)
  • Now 
  • e^{ix}=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-+... \,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
If we compare (1) and (2) we can conclude that
cos(x) + i sin(x) = e^{ix}
  • e^{-ix} = \cos(x)-i \sin(x) Hence,
  • \cos(x)=\frac{e^{ix}+e^{-ix}}{2}  & \sin(x)=\frac{e^{ix}-e^{-ix}}{2}
  • De-Moivre Theorem: (\cos(\theta)+i \sin(\theta))^n=\cos(n\theta)+i \sin(n\theta)

Complex Functions

Complex functions are mapping from complex numbers to complex numbers. Ex: f(z)=\frac{1}{z}=\frac{1}{x+iy}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}=u(x,y)+iv(x,y)

Limit of a Complex Function

Let f(z) be a complex valued function, limit l as z tends to some point z_0, there exists \epsilon, \delta, when |z\to z_0|<\epsilon, |f(z)−l|<\delta. i.e., lim_{z\to z_0}f(z)=l

Continuity of a complex function

If lim_{z\to z_0} f(z)=f(z_0) then f(z) is continuous at z=z_0
Example

Differentiablility of a complex function

Let \delta z=z−z_0, as z\to z_0 or \delta z\to 0, lim_{z\to z_0}\frac{f(z+\delta z)−f(z)}{\delta z} is the derivative of f at z_0

Analytic Functions

A complex function w=f(z) is said to be analytic at some point z_0 if \frac{dw}{dz}=f′(z)=lim_{\delta z \to z_0} \frac{f(z+\delta z)−f(z)}{\delta z} exists and unique.