14 - Complex trigonometric functions
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welcome to lecture 4 in our third week
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of the course analysis of a complex kind
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today we'll learn about the complex
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trigonometric functions so the
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extensions to the complex plane of the
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sine and cosine functions we have
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already extended the exponential
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function to the complex plane so the
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question is can we do the same thing put
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your anorectic functions now remember e
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to the I theta was our shorthand
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notation for cosine theta plus I sine
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theta therefore even minus I theta
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well you just replace theta with minus
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theta it's cosine of minus theta plus I
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times sine of minus theta but cosine is
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an even function D so just our trachea
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the real valued cosine and sine
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functions so that's cosine of theta
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but sine is an odd function so sine of
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minus theta is minus sine of theta now
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look at these two identities either the
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I theta is cosine theta plus I sine
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theta e to the minus I theta is cosine
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theta minus I sine theta if I add these
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two things up when I find is that e to
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the I theta plus e to the minus I theta
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it's simply two cosine Thetas the I sine
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theta part cancels out because it has a
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plus sign here and a minus sign there on
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the other hand if I subtract these two
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equations from each other I find that e
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to the I theta minus e to the minus I
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theta it's two I times sine theta
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because now the cosines cancel out and
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the sines don't but these two equations
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that I just obtained I can solve them
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for cosine theta and sine theta by
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simply dividing by two or two I and what
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I get is that cosine theta is e to the I
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theta plus e to the minus I theta over 2
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and sine theta is e to the I theta minus
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e to the minus I theta over two I and
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this is for the real value cosine and
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sine the idea is now well now that I can
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talk about e to the I in
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then it doesn't actually be just a theta
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I could put a Z there maybe I could just
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extend this definition to the entire
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complex plane and that's exactly what
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we're gonna do
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the complex cosine and sine functions
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are defined via cosine of Z equals e to
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the i z plus if the minus IC over 2 and
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sine Z is either the IC minus e to the
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minus IC / - I what is e to the i zi by
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the way e to the i zi suppose Z is X
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plus I Y so that's e to the i-x plus iy
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that equals e to the i-x plus I squared
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Y which is minus y so that would be e to
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the minus y which is our regular real
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valued exponential function times e to
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the i-x you could rewrite this as e to
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the minus y times cosine X plus I sine X
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and now we have expressed either the i z
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in terms of purely real valued functions
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you could do the same thing for e to the
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minus I Z and so there or cosine Z can
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be expressed in terms of all real valued
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cosine sine and exponential functions it
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will do that
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but next learn about some properties of
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the complex cosine and sine functions in
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particulate we want to compare those
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properties that we already know from
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real valued analysis to those that we
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can find for the complex equivalents
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first of all size D and cosine Z are
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analytic functions because the
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exponential function is an analytic
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function the function Z goes to I times
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Z is analytic and so you see a
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composition of those two functions right
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here we're adding another analytic
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function and then dividing by two so by
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the rules of analytic functions the
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cosine function is an analytic function
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similarly the sine function is an
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analytic function since there analytic
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in the entire plain there in fact entire
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functions for real valued Z's so if you
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plug in a zero of the form X with no I Y
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so just Z is equal to X the complex sine
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function and cosine functions agree with
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the real valued sine and cosine
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functions because in that case I get
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cosine of Z it's equal to cosine simply
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for my X plus I times 0 and that's e to
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the i-x plus e to the minus I X divided
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by 2 and that's the definition of our
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real valued cosine function what's
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cosine of minus C for the real valued
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function the cosine function is an even
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function cosine of X is equal to cosine
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of minus X is that still true for the
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complex of cosine function well cosine
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of minus C is even a minus IC plus e to
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the I Z and I can just interchange these
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two terms on top here because addition
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is commutative and I see that this is
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equal to the cosine of C sine of minus C
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is either the - I see I'm just plugging
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in a minus C for zia here
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so either the minus IC minus e to the IG
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/ - I I can't just
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jisu turns because of this negative sign
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if I do interchange these two terms I
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get an additional negative sine so sine
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minus E is minus sine of Z chuck's the
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same thing is true in real valued
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analysis sine of minus X is equal to
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minus sine of X sine function is an odd
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function remains so in the complex
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extension furthermore the addition
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properties for the sine and cosine
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functions still hold cosine of Z plus W
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equals cosine of C cosine of W minus
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sine Z son W and sine of Z plus W is
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sine Z cosine W plus cosine Z sine W how
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would you prove such a thing I'll show
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you an example how would you prove that
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cosine of Z plus W is equal to cosine Z
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cosine W minus sine Z sine W you would
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simply plug in the definitions and
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simplify let's do that let's start on
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the right-hand side
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in fact cosine Z cosine W minus sine Z
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signed up here I wrote down the
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expression for cosine Z e to the I Z
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plus e to the minus I 0 virtue the same
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thing for cosine of W minus sciency e to
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the I Z minus e to the minus i zo / - i
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and here I have sine of W I notice when
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I multiply these first two practice with
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each other the denominators multiply to
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four with the second two fractions the
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denominators are 2 I times 2 I which is
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negative 4 in the product together with
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this negative sign in front I can cancel
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the negative sign from the denominator
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and make this negative sign into a plus
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so the common denominator for these
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products of practice is 4 and the
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numerators are e to the ID z plus e to
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the minus IG it's from this term right
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there then e to the I W plus e to the
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minus I W and now I have a + remember
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from the I squared and this [ ]
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sign then I have this term and finally
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this term now when I multiply through I
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get a lot of terms we're not going to
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talk through all of these terms just
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track see these are all the terms you
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get when you multiply through for
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example e to the I Z tends to either i w
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either the I Z times e to the minus I W
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and so forth but if I look at these
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terms carefully I have an e to the I z e
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to the I W right here and there's
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another one right there
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together that gives me 2 e to the i z e
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to the IW i also happen even the - i see
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even the - i WD r and another run right
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there together that gives me two of
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these terms rest we get like the rest
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actually cancels out e to the i z times
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e to the minus i W and here I have -
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either the I Z into - I W so these two
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terms cancel out similarly even the
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minus IC e to the i w + - e to the - I
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see even I double used to these two
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terms cancel out so everything
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simplifies now I have a 2 in the
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numerator a 4 in the denominator
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I can cancel out one of those twos and
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make the denominator 2 instead and end
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up by combining the exponents with
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either Isaac's WPTZ plus W over 2 what
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that's cosine notes you post W that's
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what I wanted to show so there's no
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ingenuity in here in this proof it's
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literally plugging in the definitions
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and simplifying
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let's look at some further properties
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remember the real value cosine function
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is periodic with period 2pi is that
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still true for the complex cosine
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function let's find out what is cosine
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of Z plus 2 pi I use the definition over
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places I eat with Z plus 2 pi but what
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is either the I times Z plus 2 pi that
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is e to the i z times e to the I times 2
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pi but what is e to the 2 pi I that's
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equal to 1 so that's equal to e to the
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ID in other words this first term is
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simply e to the i z and similarly the
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second term is simply e to the minus IC
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and so I realized the cosine function is
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also periodic with period 2pi and
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similarly the sine function has the same
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period remember in real analysis sine
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squared X plus cosine squared X equals 1
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well that's still true for the new
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complex sine and cosine functions sine
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squared Z plus cosine squared Z equals 1
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how would you prove that it's actually a
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very simple proof you substitute W
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equals minus C in the addition formula
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for the cosine function let me just
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simply demonstrate how to do that the
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addition formula for the cosine function
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was cosine of Z plus W but we're gonna
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make that Z plus minus C and that's my W
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it's equal to the addition formula set
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cosine Z cosine W which is minus C for
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me minus sine Z signed up just mice
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let's simplify both sides of the
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equation ze + - e that's simply zero so
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this side of the equation becomes cosine
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of zero but cosine of zero is one on the
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right hand side cosine of minus E is the
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same as cosine of C showed that the
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cosine function is an even function so
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this is cosine Z times cosine Z which is
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cosine squared of Z on the other end
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sine of minus C is minus sine at Z so
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this minus sign can be pulled up and
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join this minus sign in order to give me
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a plus all together sine squared X so
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that's how simple it is to prove that
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formula furthermore sine of Z plus PI
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over two gives us back the cosine
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function just acid was the case for the
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real valued functions again it's very
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simple to prove look at sine of Z plus
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PI over 2 and plug Z plus PI over 2 into
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that definition of the sine function so
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wherever you see a Z make it a Z plus PI
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over 2 so sine of Z plus PI over 2 then
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equals e to the I Z plus 5 or 2 minus e
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to the minus I 3 plus PI over 2 divided
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by 2 like what is e to the I Z plus PI
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over 2 again we can pull that apart as
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we did earlier that is e to the I C
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times e to the I PI over 2 what is e to
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the I PI over 2 e to the I PI over 2
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equals I so in other words this is i
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times e to the i z similarly the second
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term is minus I come to the minus IC
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because I have an e to the minus I PI
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over 2 which is minus I this simplifies
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because I have an I in the denominator
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as well see this eye right here this I
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in the denominator cancels out with this
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eye and that I use two negative signs
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make a plus I'm left with E to the I is
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e plus e to the minus I 3 over 2 which
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is cozy
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what are the zeros of the sine function
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we know the zeros of the real valued
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sine function the real valued sine
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function looks something like this you
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know it goes to the origin and it's
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equal to zero at PI 2 pi 3 pi and so
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forth negative pi and so forth are there
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additional zeros now that we have
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extended this function to the complex
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plane that's fine now remember sine Z's
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either the is Z minus e to the minus
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side z / - I if we want that to equal to
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zero then we need the numerator of this
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fraction to be equal to zero so either
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the eysie needs to equal either - I see
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remember we proved that e to the Z is
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equal to e to the W if and only if Z
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minus W is a multiple of pi I times two
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we proved this in our lecture about the
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exponential function so e to the I Z is
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equal to e to the minus i Z even only if
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they're different in other words I see -
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I see it's not the form two K PI for
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some K and the integers but i z minus -
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sighs e that's - i Z so - I Z's needs to
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be equal to 2 K PI I I can cancel out -
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I can cancel I in this read Z needs to
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be of the form K PI where K is an
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integer
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those are exactly the zeros that we
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already knew of so the sine function
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does not gain any extra zeros by
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extending it to the complex plane it's
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never going to be 0 Alice where it's
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only 0 on the real axis similarly we can
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show that the cosine function is 0 only
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when the real value of cosine function
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was zero namely at the points PI over 2
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plus K PI where K is an integer now we
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show that the sine and cosine functions
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are analytic functions so they must have
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derivatives what are these
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the derivative of the sine function is
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simply the derivative of either isay
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minus e to the minus IC / - i - i is a
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constant that we multiply with it can go
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to the side in other words we only need
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to take the derivative of the numerator
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the numerator consists of two functions
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each of which we can differentiate
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separately in other words we need the
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derivative either the I is e and the
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derivative of e to the minus IC let's
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look at the derivative of e to the I Z
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we found the derivative in general of a
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function e to the a times C and found
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that was a time's the derivative of
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either a Z so in this case this is gonna
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be I times e to the I let's see that's
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right here the derivative of either the
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Isaac is found
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similarly we multiply with this constant
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I either the - I see when we simplify
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we see there's an I here and I here and
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I hear they all cancel out - - gives me
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a plus so I'm left with either the i zi
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plus in the minus IC / - which is cosine
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Z just like it was the case for the real
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valued sine and cosine functions the
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derivative of sine is cosine and
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similarly the derivative of cosine is
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negative sine
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finally it is possible to express the
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complex sine and cosine functions solely
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in terms of the real Sun and cosine in
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the real hyperbolic sine and cosine let
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me quickly show that to you
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sine Z is sine of X plus iy we could
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view this X plus iy X the sum of two
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complex numbers what are those complex
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numbers is X the other one is iy in
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other words we are allowed to use the
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addition formula here so this becomes
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sine X cosine iy plus cosine X sine Y
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what is cosine of Y we need to plug in I
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Y cosine formula and get e to the I
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times iy plus e to the negative I times
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pi over 2 and then back here we do the
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same thing with the sine function e to
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the I times I Y minus the minus I times
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I Y over 2 i next people begin
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simplified I times I times y that gives
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me a minus y and here I times I times y
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times negative 1 is just e to the Y and
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back here the same thing happens if I
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rewrite this I find that this first
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fraction is e to the Y plus e to the
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minus y over 2 and the second fraction I
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have an I in the denominator if I bring
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the I into the numerator 1 over I is
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equal to negative 5 so I can bring that
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into the numerator at the expense of a
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negative sign but I'm going to use that
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negative sign to flip these two terms
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with each other so I can now write e to
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the Y minus e to the minus y over 2 and
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an I here in the front but now I'm left
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with sine of X this is the hyperbolic
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cosine function cosh of y plus items
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cosine of X times sin of Y or you could
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also leave it the way it was in the
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previous line if you don't like to use
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caution
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it's inch and have another way of
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actually evaluating sine of Z in terms
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of purely real valued
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the sine of X and the exponential
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function and the cosine of X and the
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exponential function of Y and minus one
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similarly you can find that cosine of Z
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is cosine of X cosh Y minus I sine of X
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sin shui next up are some amazing
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properties of analytic functions here's
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an example if f is an analytic function
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in the domain and you write it as real
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part plus I times imaginary part would
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use the real part V is the imaginary
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part and suppose the real part U is
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constant throughout the domain then f
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itself must be constant there's no way
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to have an analytic function where only
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the imaginary part varies if the real
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part remains constant the imaginary part
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has to remain constant as well we have
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an imaginary part are not independent of
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each other they're tied together closely
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and that's what gives analytic function
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some of their amazing properties this is
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what we're gonna study in the next
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lecture