13 - The complex exponential function
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welcome to the third lecture in our
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third week of the grist analysis of a
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complex kind today we'll learn about the
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complex exponential function you started
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talking about this function at the end
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of last class we discovered that the
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function f of Z defined by e to the x
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times cosine y plus I times e to the X
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sine Y it's an entire function whereas e
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is X plus iy and e to the X cosine Y and
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sine Y are just the regular real valued
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functions that we know from calculus
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let's look at some of the properties of
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this function suppose for example that Y
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is equal to 0 so this Y right here is
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equal to 0 while cosine of 0 is equal to
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1 and so this Y would also be equal to 0
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but sine of 0 is equal to 0 so in that
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case this function would simplify the e
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to the x times 1 plus I eetu the x times
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0 but that's just either X because this
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whole last term gets cancelled out by
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the 0 so when 1 is equal to 0 then f of
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Z is simply Eve X in other words the
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function f agrees with our regular
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exponential function if you plug in
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something real it doesn't have an
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imaginary part another thing to notice
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is that we could rewrite f of Z a little
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bit we can factor out the e to the X
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term right here and are left with e to
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the x times parentheses cosine y plus I
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sine Y but cosine y plus I sine Y we
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found an abbreviation for we call that e
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to the I Y even though we weren't quite
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sure why we called it either the iy was
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an abbreviation at the time but now that
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we're looking at e to the x times e to
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the I Y we would like to think of this
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as e to the X plus iy because there
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rules for our regular real valued
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exponential function would indeed
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is to combine these two exponents we
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have improved any such thing right now
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but this is a reasonable thought to have
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and that looks like e to the Z whatever
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e to the Z might be so with these
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thoughts in mind we actually define the
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complex exponential function e to the G
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exactly like that it's also sometimes to
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notice X plus Z and it's defined by even
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the Z is e to the x times e to the I by
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where Z is X plus iy and even the iy is
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still shorthand for cosine y plus I sine
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1 this is the complex exponential
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function and we'll explored in today's
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lecture again as a reminder a to the Z
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is e to the x times e to the iy what
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does step length of a the Z so how far
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is e to the Z from the origin or the
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absolute value of e to the Z can be
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found by simply taking the absolute
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value of e to the x times e to the iy
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and we know the absolute value of the
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product of two complex numbers is the
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product of the absolute values norm of X
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norm of the iLife but e to the iy
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remember what that really is e to the I
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Y short for cosine y plus I sine Y and
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it is simply a number on the circle of
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radius one such that the angle formed
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with a positive real axis is y this is e
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to the I Y which was cosine y plus I
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sine Y all numbers of this form have
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distance 1 from the origin because
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they're on the circle of radius 1 so the
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number of e to the I Y is 1 just by how
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this number even though I wise to find
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the norm of e to the X e to the X is our
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real valued exponential function
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remember what that looks like
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e to the X looks like this this is the
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regular exponential function it's always
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above the x-axis it's always positive it
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can never be negative so the norm of e
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to the X is simply e to the X what's the
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argument of Eden is e now that we know
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how far each of the Z is from the origin
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we would like to know
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what angle is a form with a positive
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real axis what's the argument of e to
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the x times e to the I wha what is e to
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the x times e to the I Y that's a number
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in polar form where this is its length
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or radius but for a number in polar form
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radius times either I something be
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something gives to the angle let it
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forms with a positive real axis so the
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angle this number form so the positive
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real axis is y truthfully speaking I
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should have written me a plus 2 pi K
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where K is any integer so the way you
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should read this equations and argument
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of e to the Z is y what is e to the Z
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plus 2 pi I so suppose you add 2 pi I to
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you Z and then stick that into the
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exponential function while since 2 by is
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peel the imaginary that means you're
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really adding that to the imaginary part
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of Z so it's the same as e to the x
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times e to the I y plus 2 pi adding the
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2 pi to the imaginary part of Z but e to
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the I theta is a function that is
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periodic with period 2 pi because we're
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looking at the angle let us form to the
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positive real axis and it starts
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repeating after 2 pi so e to the I y
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plus 2/5 it's the same as e to the I wat
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so that this simplifies to the e the x
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times e to the iy which is it Lizzie so
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the function e to the Z is periodic and
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the period is
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to heii you add 2 pi i to Z the
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exponential function remains the same
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finally we would like to establish some
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properties that are similar to the ones
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that we have for the real valued
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exponential function we would hope that
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e to the Z plus W is e to the Z times e
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to the W let's check that this Holtz
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with our definition of e to the Z which
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is up here what is e to the Z plus W
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we'll write Z as X plus iy and W as u
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plus IV and plug that in so e to the Z
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plus W becomes e to the X plus iy plus u
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plus IV let's reorganize and collect the
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real parts and collect the imaginary
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parts because we'll need those to apply
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the definition of our exponential
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function the real part of the sum and
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the exponent is X plus U and the
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imaginary part is y plus V now we know e
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to this complex number is e to the real
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part so e to the X plus u times e to the
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I times the imaginary part so e to the I
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times y plus V that's just by definition
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of the exponential function e to the X
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plus u now we're talking all real values
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XS real use real and E's our regular
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real valued exponential function so the
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rules of why this is even the x70 of the
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you either the iy plus V we proved that
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that equals e to the iy times e to the I
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V we use the addition theorems for sine
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and cosine functions and now we just
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reorganize things again we collect e to
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the X and e to the I 1 and then e to the
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U and E to the IV since it's a product
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of complex numbers and multiplication is
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commutative we can just reorganize
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things but even the x times e to the I Y
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that's simply e to the Z and if the you
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either the IV is e to the W and we
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established indeed that the same rule
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holds for complex values as it does for
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real guys what is 1 over e to the Z Y
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climb is e to the negative Z in the
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business that when they look at e to the
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Z times even a negative see by the rule
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we just established we can simply add
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the two exponents so that's either that
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Z plus negative Cecil Z minus Z but Z
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minus Z is 0 and e to the 0 is 1 so e to
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the Z times e to the minus e is 1 now if
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we divide both sides of this equation by
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e to the Z you find that either the
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minus Z is 1 over e to the C either the
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Z is an entire function we already
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showed this that's what set us off move
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define this function what's the
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derivative an entire function has a
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derivative remember that the way the
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function e to the Z is find its real
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part is e to the X s cosine Y and its
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imaginary part is e to the x times sine
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Y again we can find these partial
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derivatives that we learned about during
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the last lecture the partial derivative
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of U with respect to X means we
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differentiate e to the X cosine Y with
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respect to X keeping in mind that Y is
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fixed in a constant the derivative of e
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to the X e to the X cosine Y is a
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constant so UX is e to the X cosine Y
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similarly V X is e to the X sine 1 uy we
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now keep X constants or e to the X is a
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constant we need to find the derivative
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of cosine wise which is minus sine Y and
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V Y is e to the X cosine Y we learned
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that for a function that's analytic its
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derivative can be found by taking UX
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plus IV X so therefore the derivative of
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our exponential function is e to the X
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cosine Y which is UX plus I times e to
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the X sine Y that's the VX
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but that's e to the Z in other words the
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derivative of e to the Z is e to the Z
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again or in singles DBC of e to the Z is
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e to the Z notice that we're using not
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the partial derivative symbols which
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were GDX we found we use that notation
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for the partial derivatives this is the
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complex derivative we use the way Euler
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d for that notation so similarly to the
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real-valued exponential function the
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derivative of the exponential function
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is the exponential function itself
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similarly the derivative of e to the
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constant times e how do you find that
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this is actually the composition of two
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functions the exponential function
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composed with a function that multiplies
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Z by a in other words we need to find
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the derivative of the outside function
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times the derivative of the inside
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function the inside function is a time Z
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its derivative is simply 8 therefore we
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get a times either the AC by the chain
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rule for the derivative of e of the AC
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what is e to the complex conjugate of C
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the complex conjugate of Z is X minus I
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Y Z was X plus iy therefore e to the Z
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conjugate equals e to the X is e to the
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minus I Y but we showed earlier and we
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can remember this in the picture what is
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e to the minus iy even the minus iy
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means for taking a negative angle minus
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y and get a number down here if we
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reflected that with respect to the x
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axis we would get this number up here so
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this is e to the minus I Y and this is
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either the iy in their complex
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conjugates of each other as this picture
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shows in other words even the minus iy
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is the conjugate of e to the iy e to the
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IX is real value we might just as well
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put a conjugate on top of it it makes no
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difference because real number equals
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its complex conjugate so we can extend
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the conjugation over the entire product
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but that's your conjugate of a Divisie
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in other words e to the conjugate of C
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equals the conjugate of e to the c1 is e
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to the Z equal to 1 well we need either
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the x times e to the iy to be 1 this is
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a number in polar form when it's a
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number in polar form equal to 1
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well it's length needs to be 1 it's
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angling to be 0 or 2 pi or 4 pi so in
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other words we need either the X to
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equal 1
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and we need Y to be zero or 2pi or 4pi
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when it's either the x equal to one
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that's the case when X is equal to zero
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so we need X is zero and Y of the form
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two K pi in other words we need Z to be
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of the form X plus iy so zero plus I
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times two K pi so those are the points
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where the exponential function takes the
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value 1 1 is e to the Z equal to e to
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the W we already saw that if Z is equal
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to W plus 2 pi I then e to the Z equals
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e to the W but are there other
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possibilities for e to the Z to equal e
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to the W let's find out if we divide
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both sides of this equation by e to the
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W we know that 1 over e to the W is e to
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the minus W so either the Z divided by
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either the W becomes e to the Z minus W
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and that then needs to be equal to e to
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the W over e to the W which is 1 when
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this e to the Z minus W equal to 1 well
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that is the case when Z minus W is of
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the form 2 pi K in other words when Z is
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equal to W plus a multiple of 2 pi I so
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the cases we observed earlier were the
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only ways for e to the Z to equal e to
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the W let's understand that mapping w
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equals e to the Z we did this earlier
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for some other functions we look at it
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as a mapping from the Z plane to the W
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plane with W being equal to e to the C
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for example let's look at images of
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horizontal lines what's a horizontal
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line a horizontal line is of the form X
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plus iy 0 where y 0 is fix we're fixing
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the imaginary part and just varying X
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those points X plus iy 0 or they get
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mapped to e to the X plus iy 0 which is
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e to the X e to the iy 0 but Y zero is a
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fixed number so this is not going to
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change this is fixed and the only thing
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that can change is e to the X and e of X
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if X runs from negative infinity to
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infinity either the X will go from 0 to
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infinity it will never take the value 0
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but anything as close to zero as
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possible and even though iy is a fixed
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angle from the origin let's look at some
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examples for example this red dotted
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line right there
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what is its image it's the line where y
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0 is equal to PI what is e to the I pi e
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to the I pi is equal to negative month
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performing an angle of Pi with a
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positive real axis in other words all
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these numbers either the x times e to
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the i y0 lie on the negative real axis
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so the image of the red dotted line
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right here is the negative real axis
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accepting the origin
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let's look at the green line next what
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is its image in my picture it looks like
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the imaginary part is a little bit
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bigger than PI over two this angle here
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is just a little bit bigger than PI over
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2 e to the iy 0 is a point on the unit
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circle whose radius forms the angle y 0
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with a positive real axis and therefore
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the image of the green line is this
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green line right here let's look at the
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turquoise line
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it looks like it's imaginary part is
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around PI over 4 so the image is going
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to be this turquoise line right there
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how about the yellow line the yellow
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line up here has an imaginary part just
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a little bit bigger than pi so looking
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at this angle right here and therefore
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the image of the yellow line is this
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radial segment down here you notice that
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I drew another yellow line in the Z
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plane a little bit above minus I PI and
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indeed this reflects that the
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exponential function is periodic with
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period 2 pi I this second be on a line
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has distance 2 pi I from the other
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yellow line in other words it's image is
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the same as the image of the other
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yellow line so these two yellow lines
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both get mapped to the radio segment
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over in the W plane finally there's a
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purple line here and you can check that
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it gets them up to something like this
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purple line over here what are the
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images of vertical lines a vertical line
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is of the form X 0 plus iy where now X 0
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is fixed so the real part is fixed in
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the imaginary part varies those points
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get mapped to either X 0 plus iy which
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is feed the X 0 times e to the I 1 so
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now this number is fixed the distance
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from the origin is fixed for all these
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images and the angle varies in other
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words for a fixed line that's vertical
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its image is a circle for example the
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yellow line that is on the imaginary
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axis so X 0 will be 0 it's mapped to
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either 0 just 1 times either that I want
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but e to the I Y if Y is allowed to vary
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from negative infinity to infinity gets
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mapped to the circle and the circle is
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mapped over and over and over
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again because this keeps repeating
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there's zero here and say 2pi is right
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here then this little short segment
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already gets mapped to the full unit
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circle and afterwards we just keep
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repeating
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similarly the red dotted line gets
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mapped to e to the x 0 times e bi y this
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x0 is a little bigger than 0 so either
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the x0 is bigger than 1 so it gets
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mapped to a circle whose radius is
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bigger than the previous one and if we
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move over to the green line its radius
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it's going to be even bigger than the
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one of the red circle similarly if I go
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to negative x zeroes e to the something
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negative is going to be a number less
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than what it's so positive but a number
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less than 1 so these radii are going to
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be less than 1 and the circles are going
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to be sitting inside the unit circle
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putting it all together we can actually
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figure out what the image of a vertical
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strip is suppose we look at the set of
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all Z whose real part is between zero
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and one so here's one and there's zero
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and this is my vertical strip s what is
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its image we already saw that the yellow
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line which is on the imaginary axis gets
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mapped to the unit circle the green line
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for which I said the real part is equal
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to 1 gets mapped to the circle of radius
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II in other words the strip from 0 to 1
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gets mapped to this annulus between the
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circles of radius 1 and of radius II if
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I draw a horizontal line this picture in
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addition the one that I drew is maybe at
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PI over 4 then I cut out a portion of
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that annulus so for example this
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turquoise portion here of the strip it's
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mapped to this turquoise portion of the
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annulus these kinds of pictures help you
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understand what the exponential function
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does graphically there's a few final
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questions when is e to the Z equal to 0
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well find out e to the Z is equal to 0
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when e to the x times e to the iy is
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here we know e to the I Y those are all
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numbers on the circle of radius 1 they
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can never be 0 when is a product of two
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numbers 0 well one of the factors has to
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be 0 the second one can't be 0 so it has
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to be the first one so we need to either
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bags to be 0 remember what even the X
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looks like either the X looks like this
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it never crosses the x axis it's never
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equal to 0 in other words either the Z
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can never take the value 0 just like the
22:34
real valued exponential function it
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never takes the value 0
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let's go a little bit further suppose
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I'm giving you a point in the complex
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plane not the origin but any other point
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then cannot find a W such that e to the
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W is equal to that point Z let's write Z
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in polar form as absolute value of Z
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tendency to beg theta and that's right W
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in Cartesian form because that works
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best for this problem so that's right W
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is U plus IV then e to the W being e to
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the Z is the same as e to the U times e
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to the IV which is feed of the W being
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equal to absolute value of Z that I say
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that which is simply Z and this was e to
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the W but these had two numbers in polar
23:33
form when are two complex numbers in
23:35
polar form the same well they would have
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to agree in their distance from the
23:40
origin so even if you would have to be
23:42
the absolute value of Z and then their
23:44
angles have to agree modulo 2 pi e to
23:46
the IV would have to be eaten by theta
23:48
in other words when is e to the u equal
23:53
to the absolute value of Z this is all
23:55
via values now that's the case when use
23:57
a natural logarithm of the absolute
23:59
value of Z I'm going to e to the IV the
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same as e to the I theta well being
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theta can only differ by a multiple of 2
24:08
pi
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in other words w2c plus IB has to be of
24:14
the form natural log of Z that's my U
24:16
plus I times an argument of Z which is
24:20
what theta is you just saw the basic
24:25
ideas and how to define a complex
24:27
logarithm you see there's an issue
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already with the argument being here the
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argument function is a multivalued
24:34
function and we'd have to decide how to
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pick the argument so that we could
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hopefully make the logarithm function an
24:39
analytic function and indeed that as
24:41
possible