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Complex numbers - Exercises with detailed solutions 1. Compute real and imaginary part of z = i¡4 2i¡3: 2. Compute the absolute value and the conjugate of z = (1+ i)6; w = i17: 3. Write in the \algebraic" form (a+ib) the following complex numbers z = i5 +i+1; w = (3+3i)8: 4. Write in the \trigonometric" form (‰(cosµ +isinµ)) the following ...
The problems covered include finding the general value of log(1+i)+log(1-i), evaluating logarithms of expressions involving trigonometric functions, and using properties of logarithms to simplify complex logarithmic expressions.
Examples for Complex numbers Question (01) (i) Find the real values of x and y such that (1 ) 2 (2 3 ) 3 3 i x i i y i i i i − + + + + =− − + (ii) Find the real values of x and y are the complex numbers 3−ix y2 and − − −x y i2 4 conjugate of each other. (iii) Find the square roots of 4 4+i (iv) Find the complex number Z satisfying ...
14 Σεπ 2017 · The complex logarithm. If the logarithm and the exponential function are to be inverses, then. elog(z) = z = rei. = eln(r)+i. so the only possible de nition for the complex logarithm is. log(z) = ln jzj + i arg(z): Problem: There are in nitely many values for arg(z).
Chapter 13: Complex Numbers. Sections 13.5, 13.6 & 13.7. 1. Complex exponential. The exponential of a complex number z. x = + iy. is defined as. exp(z ) = exp(x + iy ) = exp(x ) exp(iy ) = exp(x ) (cos(y ) + i sin(y )) As for real numbers, the exponential function is equal to its derivative, i.e.
Complex logarithm Complex power function De–nition Properties 1. Complex exponential The exponential of a complex number z = x +iy is de–ned as exp(z) = exp(x +iy) = exp(x)exp(iy) = exp(x)(cos(y)+i sin(y)): As for real numbers, the exponential function is equal to its derivative, i.e. d dz exp(z) = exp(z): (1) The exponential is therefore ...
Complex logarithms. To solve ew = z , for z 6= 0 : Let w = u + iv ; so ew = eueiv ; and write z = jzj e i : eueiv = jzj e i , eu = jzj ; eiv = ei : Infinitely many solutions w : u = log jzj ; v = + 2 k. Describe all solutions by: log(z) = log jzj + i arg(z) Examples: log(2i) = log(2) + i. 2 + i2 k. log( 3) = log(3) + i. + i2 k. p.
