Looking for a Tutor Near You?

Post Learning Requirement »
x

Choose Country Code

x

Direction

x

Ask a Question

x

x
x
For better view click here
x
Hire a Tutor
DISCOVER

Simple And Esy Notes On Maths And IC Engine

Loading...

Published in: Mathematics
1,019 Views

INFINITE SERIES,LINEAR ALGEBRA,Linear Differential Equations,ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER,COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLE,LAPLACE TRANSFORM,IC ENGINE MECHANISM,MULTI POINT FUEL INJECTION,PISTON INFORMATION,Testing of IC Engines

Prashaant / Kolhapur

7 years of teaching experience

Qualification: B.E.(Mechanical Design)

Teaches: Algebra, Chemistry, English, Mathematics, Physics, AIEEE, GATE Exam, Polytechnic Entrance, Drawing, Mechanical, Production, Robotics

Contact this Tutor
  1. COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLE Definition of a complex number : A number of the form x+iY , where x and y are real numbers and i = ,is called a complex number, The set of complex numbers is denoted by C. If z = x+iY is a complex nmber ,then x is called the real part of z and we write Re(z) = x. y is called the imaginary part of z and we write Im(z) —y. If x—0 and ,then z =0+iY —i y is called purely imaginary number. If y=0 ,then Z=X+i0 =x is a real number. If x=0 and y=0 ,then z= 0+i0 is called the zero complex number. Conjugate of a complex number : If z= x+iY then = x-iY is called conjugate of z. Properties: 2) z+w=z+u.' 5) If 0, then w : c 6) z is real' Properties of complex numbers : (a) The sum,difference, product and quotient of two complex numbers is a complex number. (b) If a complex number is zero , then its real and imaginary parts are separately equals to zero. Thus x+iY so x=0 and y=0 (c) If two complex numbers are equal ,then their real and imaginary parts are separately equal.Thus x+iY =a+ib so x=a and y=b (d) If two complex numbers are equal ,then their conjugates are also equal. Thus x+iY =a+ib so x-iY —a-ib Polar-form of a Complex number: • Polar Representation of complex numbers z = x + iy = r(cos0 + i sin 0) , where the real number r = Izl (called the absolute value (or the norm, or the modulus) is defined by
  2. Izl 2 and the angle 0 (called the argument) is defined by x and sine = cose = The argument arg z can be thought as an infinite set of values arg z = {0+ 2nnIne 2 Z} •. For any complex numbers zl and z2 there holds rl(cos 61 + i + i sin62) = + i sin(61+62)] Iz1z21 = Izll Iz21 In other words, arg(z1z2) = arg zl + arg z2 (mod 271) . Note: The Arg(z) is the angle B, and that this angle is only unique between called the primary angle. Adding 27 If Zl and zz are two complex numbers then —T O # which is (a) (b) (c) (d) (f) Iz1z21 = Izll Iz21 arg zl + arg z2 arg(Z2) =argzl -argz2 IZI +Z21 < I + 1 IZI -Z21 > -Z2 Iz12 2 = zz argz+argä = 0 DeMoivre's Theorem: DeMoivre's Theorem is a generalized formula to compute powers of a complex number in it's polar form. r(cosO+i from the eariler formula we can find (z)(z) easily: Looking at Then; = = j" 2 (COS20 + i Sin ( •-2 vs (eos30 + isin 30) Which brings us to DeMoivre's Theorem: — r(cosO + i and n are positive integers then — v" (cosnO+ sin no ) If Basically, in order to find the nth power of a complex number we take the nth power of the absolute value or length and multiply the argument by n.
  3. Statement of DeMoivre's Theorem (a)lf n is any integer, positive or negative,then + i sin0)71 = Cosne+i Sinne (b) If n is a fraction,positive or negative ,then one of the values of (Coso + i SinO)n = Cosne+i Sinne Example : Is (Sino + i Cos0)Tl = Sin né+ i Cos ne (Sino + i cos0)n Sin ne+ i cos ne Solution : Because, n = [COS Example : Simplify ( (Sino + i Cos0) _ + isin cos(n— — 710) + i sin(n— — 710) 2 2 Sin né+ i Cos ne n COSO+iSinO 4 SinO+iCOSO Solution : COSO+iSine 4 SinO+iCOSO COSO+iSinO cose-e +iSiné-O) (COS9+iSinO)4 (COSO+iSinO)4 (COSO+iSinO)4 COS40-iSin40 (COSO+iSinO)4 (COSO+iSinO)-4 = (cose + isin0)8 =/cos89+i sin89

Need a Tutor or Coaching Class?

Post an enquiry and get instant responses from qualified and experienced tutors.

Post Requirement

Related Study Notes

Upload Study Notes

If you have your own Study Notes which you think can benefit others, please upload on LearnPick. For each approved Study Note you will get 100 Credit Points and 100 Activity Score which will increase your profile visibility.

Upload Now
Home > Study Notes > Simple And Esy Notes On Maths And IC Engine