Elementary Complex Functions

Dr Viktor Fedun

v.fedun@sheffield.ac.uk

Instantaneous velocity

Let us consider one dimensional motion of some physical object B.
The plot above represents a path of this object as a function of time.
The instantaneous velocity at the time moment t0 is the derivative of x with respect to time at the time moment t0.

In the vicinity of point x0 we can approximate motion of B by motion with constant velocity:

In that case trajectory will be approximated by a tangent line. Such an approximation will not be accurate outside small neighbourhood of time moment t0.

Approximation of motion

We can approve our approximation if we will take acceleration onto account. Acceleration is the second derivative of displacement with respect to time:.

In the vicinity of point x0 we can approximate motion of B by motion with constant acceleration using well known equation:

In that approximation will be more accurate but still deviate from real trajectory after some time,

Approximation of motion

Approximation of motion



In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.



The notation n! was introduced by French mathematician Christian Kramp in 1808.





Taylor Series





If t0=0



In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715.

If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin.

Taylor Series: exponential function







Taylor Series: sine







Taylor Series: cos







Taylor Series



A Taylor series approximates the chosen function.

Taylor Series



As we increase the number of terms, the Taylor polynomial for the function gets closer to the function. The function can be thought of as a polynomial of infinite degree.

Taylor Series




Understanding of Negative and Complex Numbers

the equation x2 = 9 really means:






What transformation x, when applied twice, turns 1 to 9?

The two answers are x = 3 and x = -3:

That is, you can “scale by” 3 or “scale by -3 and flip”

(flipping or taking the opposite is one interpretation of multiplying by a negative).


Now let’s think about x2 = -1, which is really






What transformation x, when applied twice, turns 1 into -1?

  • We can’t multiply by a positive twice, because the result stays positive
  • We can’t multiply by a negative twice, because the result will flip back to positive on the second multiplication



  • But what about… a rotation! It sounds crazy, but if we imagine x being a “rotation of 90 degrees”, then applying x twice will be a 180 degree rotation, or a flip from 1 to -1!





    And if we think about it more, we could rotate twice in the other direction (clockwise) to turn 1 into -1.
    This is “negative” rotation or a multiplication by -i:




    If we multiply by -i twice, we turn 1 into -i, and -i into -1.
    So there’s really two square roots of -1: i and -i.

  • i is a “new imaginary dimension” to measure a number;
  • i. (or -i) is what numbers “become” when rotated;
  • Multiplying i is a rotation by 90 degrees counter-clockwise;
  • Multiplying by -i is a rotation of 90 degrees clockwise;
  • Two rotations in either direction is -1: it brings us back into the “regular”; dimensions of positive and negative numbers.


  • Let's go further:

  • 1=1 (No questions here)
  • i=i (Can’t do much)
  • i2=-1 (That’s what i is all about)
  • i3=(i*i)*i=-1*i=-i(3 rotations counter-clockwise = 1 rotation clockwise.)
  • i4=(i*i)*(i*i)=-1*-1=1(4 rotations bring us “full circle”)
  • i5=i4*i=1*i=i(Here we go again…)



  • Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions “X” and “Y”. Or anything with a cyclic, circular relationship.




    Roots of Unity




    Can a number be both “real” and “imaginary”?







    In fact, we can pick any combination of real and imaginary numbers and make a triangle. The angle becomes the “angle of rotation”. A complex number is the fancy name for numbers with both real and imaginary parts. They’re written a + bi, where


  • a is the real part
  • b is the imaginary part












  • Trigonometry and complex numbers




    Each point P in the plane may also be identified by a pair of numbers r,
     where r is the distance from P to the origin,  OP, and θ is the angle from the positive x-axis to OP.
    <> Since θ is in standard position, the coordinates of any point on the terminal side are expressible as (r cos(θ), r sin(θ)).
    Thus if P is identified with the complex number x+yi, then P has coordinates  x, y







    We may also write the complex number x+yi as






    The form r(cos(θ)+i sin(θ)) is called
    the trigonometric or polar form of x+yi.

    The nonnegative number r is called the absolute value or modulus, and θ is the argument of the complex number.




    Exponential function

    How ex is defined in elementary calculus?



    a) As unique solution to the differential equation:






    b) By the power series:






    Exponential function

    Definition of ez for imaginary z=iy






    Expanding and regrouping the terms:






    Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number y






    Exponential function

    Definition of ez for complex z=x+iy



    To define ez for complex number z=x+iy
    we use the property of real exponent ea+b = ea eb

    Thus ex+iy =ex eiy


    Definition:

    If z=x+iy, then ez is defined by:






    Exponential function

    Geometrical illustration of eiy



    Polar form:






    or






    Exponential function

    Examples



    Rewrite the following expressions in the form a+ib :e4+2i and e-3-2i









    Trigonometric functions

    Definition of cos(z) and sin(z) for complex z



    We know now that






    In these equations y is some real number.
    We already know how to calculate e3i but we still do not know the meaning of sin(3i)?



    Trigonometric functions

    Definition of cos(z) and sin(z) for complex z



    We know now that






    Thus






    Trigonometric functions

    Definition of cos(z) and sin(z) for complex z



    Definition: If z=x+iy, then cos(z) and sin(z) are defined by:






    Trigonometric functions

    Example



    Express in the form a+ib: cos(1+2i)






    De Moivre's theorem

    From




    a complex number z may be expressed in terms of its modulus |z|=r and argument arg(z)=θ in the exponential form:





    Using the property





    of the exponential function, we have, for any n,





    so that





    This result is known as de Moivre's theorem.






    Abraham de Moivre (1667 France – 1754 London, age 87)



    Example 1









    Example 2







    Example 3



    Roots of a Complex Number

    Examples and Solutions