And people don't know what it means graphically to input a complex angle in a cosine function too, so the best it can be understood is the Taylor expansion of cosine.
Symmetrical, Ok! And showing it graphically on the complex plane, with a red-pen-black-pen would be challenging. I prefer the insight through the Taylor series.
Hi Professor, Can you explain to me why these definitions are valid and not just circular reasoning, since the complex exponential is defined in terms of sin and cos to begin with? Thanks!
Aaron Altman sort of, but by that standard all math is circular. He's taking the constructed relation and rearranging the form to isolate the trig functions to find their simplest complex expressions.
The way I see it, he's showing what happens when you sub in a complex number to the e^iΘ formula. It's just subbing in and re-arranging to find another way of expressing an idea. AIUI there are genuine deep questions about circular reasoning in maths I'm not sure it's immediately relevant here. FWIW I have the same problem with trig identities. To me they are just going round in circles, but maybe that's the point - they're not meant to be referencing anything "external".
Aaron Altman it's a proof of the sin and cos definitions, not of the Euler formula , that is proven WITHOUT using those definitions. So it's not circular. The use of the word " definition" is problematic, hence your question. By the way is super easy to show that, with those "definitions" the pitagorean formulas are preserved
Would you believe me if I told you that I was just using Euler's Formula and Demoivre's theorem to sum the series for sin(rx) (r is the summation variable) and I got to the part where I needed to simplify the complex fraction... And I was just about to do a google search on these definitions and my phone shows me a notification. Clicked on it and here I am! Blackpenredpen be the best❤
Oh wow, I never thought about trying to solve for the cos and sin from euler's formula. This video was very helpful. Is there any intuitive way to see how/why the hyperbolic functions came into existence? As DougCube posted a little bit ago, cosh(z)=cos(z/i), but this just seems like mathematical hocus pocus. Does it have to do with the definition of x^i as being a rotation around the complex plane? Also, on that note, is this identify effectively the axiom of how to interpret a^i? Or does some other base axiom lead to a^i needing to be a rotation around the complex plane? These two questions may be related, I'm not sure.
@@РусланМерзляков-х9б (a^i)^i = -a The first i rotates it from (1)a to (i)a, and the second i rotates it back around to (i)(i)a = (-1)a This is, I believe, a function of how i is defined. What I don't really know what is why rotations in the complex plane have anything to do with square roots of negative numbers,
Good video and what a great job you have did!!! I wish that many people could watch this video,very helpful.I have a suggestion:We can use this definition to solve for ∫1/(1+tanx) dx [This is the integral I got after simplifying ∫ √cotx/(√cotx+√tanx) dx from Oon Han]from 0 to π/2.
I have a question about Hyperbolic functions, i know that cosh²(x)-sinh²(x)=1, how from that they got to the definition of cosh and sinh. how someone came with the idea to define those 2 the way they are?
Now I don't have to remember all of those Trig Integral Identities! jk haha Also, your channel is really cool, dude. Just found it, and you do a bunch of cool stuff. You should do some nonlinear differential equations, those are really cool.
Aren't functions with real domains extended to the set of complex numbers using Analytic Continuation? (I've only heard of the name) Is using Euler's proposition a valid method to derive that?
Will i ask a question of if we have 1-e to power negative i L alpha upon 1-e to power negative i alpha how can proved sin{L+1/2} alpha upon sin(alpha/2)
Why do you multiply the bottom equation by -1 before adding it to the top equation (two steps) when you can also just subtract the bottom equation from the top equation (one step)?
Subtraction is only just addition with a (-1*x) in front, just like division is really just multiplication with (1/x) instead. They are inverses, but using the full definition of the inverse is helpful, especially when doing weird stuff like add equations.
kindlin My point was that I don't think it's helpful. People who understand that you can add equations will also understand that you can subtract them. Explicitly multiplying with -1 is just a needless distraction imo.
Well, he's always quite explicit, and I don't think adding equations is quite as straightforward as basic algebra, but that's my opinion. I like his explicitness. Whenever I see entire equations being manipulated like that I always have to think it through carefully, because it's not just inverse this or add that, entire parts of the equations seem to vanish.
Hey, "master of math". Did you forget to mention that z is not a complex number itself? Meaning that your formula proof works ONLY when z is real? How about cos(x+iy) ? You forgot that you deal with complex numbers? How come?
Yes but to do that you need the find out the argument(the angle between th number and the x axis) of the number for which you have to use inverse trig functions of real variables -1 to 1 So you cant cheat with it.
I'm not so sure about the first step because exp(z)=exp(x)*(cos(y)+i*sin(y)) by definition. could you please clarify why we are allowed to leave out the exp(x) ?
Define them in terms of the lambert W function and you'll win the 8th millennial prize. It's a secret btw. The first 7 are just to distract you xD Well played willy wonka, well played.
Hey there! Great fan of yours. But here lies a slight CONCEPTUAL ERROR in your solution to this! EULER THEOREM is only valid for REAL inputs / numbers You have taken (z) as a complex number and hence cannot be applied in the theorem. Thus making the solution COMPLETELY WRONG!!!! Please check on it once and reply ASAP This was a humble notification ( no offences) ☺️
There is not one way to "get" Euler's formula or, for that matter, to define sine and cosine. There are multiple equivalent definitions that can be inter-proved with each other. The most intuitive definition is that sine and cosine coordinatize the unit circle with a uniform speed parameterization, but this does not really help in extending to complex numbers - the so-called "analytical extension". Taylor series provide the analytic extension; so it it is better to first develop them from the real number plane definitions of sine and cosine, then use them as definitions of the _complex_ sine and cosine functions. You could also use the differential equations: df/dz = f, f(0) = 1 does also work on the complex numbers to define the exponential, and d^2 f/dz^2 = -f, f(0) = 0 does also define the complex sine (and if f(0) = 1, complex cosine. But this I think may require even more complex analysis to prove the complex analogues of the differential equation rules work, than just the basics with analytical extensions via Taylor series. The limit exp(z) = lim_{n->oo} (1 + z/n)^(1/n) also well serves to define exp on the complex plane just as much as it does for real numbers, and you can prove that exp(iz) = cos(z) + i sin(z) without even needing an analytical extension for cosine and sine beyond their real-number definitions as just mentioned.
The only TH-camr who actually listens to his comments time to time. Great job as usual 😊
Thanks!!!
I try!!! And will even try harder after your comment!!
Cheers!
bprp
blackpenredpen Glad to hear that!
+
Now let's use the formula to do some crazy integrals.
trucid2 Like fourier transforms?
Nikola Anderbaum let's make everything lead to QM
that why im here .lol
it's all understandable, until the implication came
I really like the excitement for math that you exude in each video. Your enthusiasm is contagious!
you could also explain that Cos(z)=Cos(-z) because is symmetrical, I think it is more common to know the graphic than the Taylor series.
Daniel Alejandro Acevedo Castaño
Well, people who don't know Taylor series can't understand the euler's formula, thus this whole video
And people don't know what it means graphically to input a complex angle in a cosine function too, so the best it can be understood is the Taylor expansion of cosine.
mjtsquared yes but that complex angle wouldnt change the fact that the function in symmetrical. Thus, he is correct.
Symmetrical, Ok! And showing it graphically on the complex plane, with a red-pen-black-pen would be challenging. I prefer the insight through the Taylor series.
Cosine is an even function. That's one of its most fundamental properties. You kinda have to know that. But it's nice to know where it comes from.
The master of mathematics! Thank you seriously for this demonstration!
A great classic! Always a pleasure to see :D
Yay! Thank you!!
Here's a cool idea: You can use this definition to integratre (sin(x))^n and (cos(x))^n via the binomial theorem.
You can move i from the denominator, and you would end up with the equation
sin(z) = -i(e^(iz) - e^(-iz)/2) => sin(z) = -i * sinh(iz)
yep, sin and sinh are (essentially) 90 degree rotated copies of each other
Now I zee, thank you.
💗💗💗 nice explanation sir thank u sir
I wonder if it's possible to graph sin(x) using its complex definition.
Whoa! Man you just opened all the windows of my brain. Now i can do really cool stuff with this. Thanks 😁
Another wonderful video. Thank you Professor.
I usually don't comment on videos, But this one, you exactly brought what I needed. Thank you
Hi Professor,
Can you explain to me why these definitions are valid and not just circular reasoning, since the complex exponential is defined in terms of sin and cos to begin with? Thanks!
Aaron Altman sort of, but by that standard all math is circular. He's taking the constructed relation and rearranging the form to isolate the trig functions to find their simplest complex expressions.
The way I see it, he's showing what happens when you sub in a complex number to the e^iΘ formula. It's just subbing in and re-arranging to find another way of expressing an idea.
AIUI there are genuine deep questions about circular reasoning in maths I'm not sure it's immediately relevant here.
FWIW I have the same problem with trig identities. To me they are just going round in circles, but maybe that's the point - they're not meant to be referencing anything "external".
Aaron Altman it's a proof of the sin and cos definitions, not of the Euler formula , that is proven WITHOUT using those definitions. So it's not circular. The use of the word " definition" is problematic, hence your question. By the way is super easy to show that, with those "definitions" the pitagorean formulas are preserved
Math is just manipulation of symbols within a formal system. If it was linked to anything external it wouldn't be math.
Aaron! How's school going?! I will go find u one day and film videos! lol
Amazing, straight to the point, world class teaching
Would you believe me if I told you that I was just using Euler's Formula and Demoivre's theorem to sum the series for sin(rx) (r is the summation variable) and I got to the part where I needed to simplify the complex fraction... And I was just about to do a google search on these definitions and my phone shows me a notification. Clicked on it and here I am! Blackpenredpen be the best❤
You are great, can you slow down, just a bit? It was worth watching twice, this is really beautiful. I hope you keep these coming.
Oh wow, I never thought about trying to solve for the cos and sin from euler's formula. This video was very helpful.
Is there any intuitive way to see how/why the hyperbolic functions came into existence? As DougCube posted a little bit ago, cosh(z)=cos(z/i), but this just seems like mathematical hocus pocus. Does it have to do with the definition of x^i as being a rotation around the complex plane?
Also, on that note, is this identify effectively the axiom of how to interpret a^i? Or does some other base axiom lead to a^i needing to be a rotation around the complex plane? These two questions may be related, I'm not sure.
I also think that a^i should not be circuit because a^(-1) isn't circuit and (a^i)^i should be equal a^(i^2)=a^(-1)
@@РусланМерзляков-х9б
(a^i)^i = -a
The first i rotates it from (1)a to (i)a, and the second i rotates it back around to (i)(i)a = (-1)a
This is, I believe, a function of how i is defined. What I don't really know what is why rotations in the complex plane have anything to do with square roots of negative numbers,
why is this unlisted?
Random Tutorials it says uploaded today.......
Daniel Newville And the odd thing is that the comment of Random Tutorials is 2 days, while the vid has been published some hours ago.
nope
I love your personality! Please never stop making videos! :D
can you show solving some trig integrals using these definitions?
nathanisbored I did. Check out integral of sec(x),4 results
thanks
Good video and what a great job you have did!!! I wish that many people could watch this video,very helpful.I have a suggestion:We can use this definition to solve for ∫1/(1+tanx) dx [This is the integral I got after simplifying ∫ √cotx/(√cotx+√tanx) dx from Oon Han]from 0 to π/2.
I went into Fourier series not knowing this, this cleared a lot of stuff, thanks.
I have a question about Hyperbolic functions, i know that cosh²(x)-sinh²(x)=1, how from that they got to the definition of cosh and sinh. how someone came with the idea to define those 2 the way they are?
Great work ! Really thanks .
Thank you so much. We are doing this in school and this video makes it make so much sense.
You are a great person 😙😙😙
can i get a some sources on why we put z and -z.like the thought process or the reason
Bro is the GOAT.
Excellent explanation, and easy to derive and remember.
for sinx is (just a little) more easy to replace cosz=(eiz+e-iz)/2 in cosz+isinz=eiz
Thanks, man. Good Job as usual. It seems you only make videos on math courses' tricks. By the way, are you chinese?🙂
Can you express inverse sine and cosine in terms of e?
Can you do it for the other trig functions.
very professional thanks for the help
Why is there an i in the denominator? Wouldn’t you usually realise the fraction?
Now I don't have to remember all of those Trig Integral Identities! jk haha
Also, your channel is really cool, dude. Just found it, and you do a bunch of cool stuff.
You should do some nonlinear differential equations, those are really cool.
Very good explanation, and video as well ! Thank you so much
Thank you Sir
Is there any chance you could do videos on real analysis? Like specifically metric spaces and sequences/series in metric spaces
you always blow my mind up !!!
Can you explain where do these Cosh, Sinh, and whatever do?
thanks, more complex analysis please
Thanks man. It was amazing and helpful for me
What would be the inverse function of sine(i)?
you saved my day sir tanks a lot
Let's check out what cosh(ix) & sinh(ix) are!
Aren't functions with real domains extended to the set of complex numbers using Analytic Continuation? (I've only heard of the name)
Is using Euler's proposition a valid method to derive that?
But in e^iθ, θ is a real, i.e. the argument of a complex. Is it valid to extend cosine and sine to any complex z?
Parabéns, muito bom!
I fucking love you thanks for helping me with College
Thanks ALOT, Very helpfull.
Will i ask a question of if we have
1-e to power negative i L alpha upon
1-e to power negative i alpha how can proved sin{L+1/2} alpha upon sin(alpha/2)
What is range of sinz?
This is great. Thanks so much.
Would this work for the real numbers as well, using complex numbers to define a real cosine and sine? That is, cos(x) = (e^ix + e^-ix) / 2 , et. al.?
thanks for your video
How do we know that cos(z) = cos(-z)???
because here z is complex
Hi sir best evidence
Why do you multiply the bottom equation by -1 before adding it to the top equation (two steps) when you can also just subtract the bottom equation from the top equation (one step)?
It's the same. It's just more detailed
Subtraction is only just addition with a (-1*x) in front, just like division is really just multiplication with (1/x) instead. They are inverses, but using the full definition of the inverse is helpful, especially when doing weird stuff like add equations.
kindlin My point was that I don't think it's helpful. People who understand that you can add equations will also understand that you can subtract them. Explicitly multiplying with -1 is just a needless distraction imo.
Well, he's always quite explicit, and I don't think adding equations is quite as straightforward as basic algebra, but that's my opinion. I like his explicitness. Whenever I see entire equations being manipulated like that I always have to think it through carefully, because it's not just inverse this or add that, entire parts of the equations seem to vanish.
What is the relation of Euler Number with the Complex Numbers ?
Guilherme Guimarães
Taylor series
is there a non-complex algebraic definition of sin/cos?
Hmm, we can do Taylor series
@@blackpenredpen ok...i was looking for the way calculators compute it. are they using taylor series or this complex definition?
I got backlog in maths and watching this now to prepare for supplementary.. lol
Thanks, doing great man!
Why are we allowed to substitute theta with a complex number, z?
Eulers formula is valid for complex theta as well. exp(z) is defined as a power series.
Kuratius I haven't learnt/thought of exp(z) expressed as a power series!! I'll go read up. Thanks!
Amigo buenas tardes cómo sería la
Solución de e^ix . e^iy???
You can't plugin z into Euler's formula, because Euler's formula require that theta to be a real number
Euler doesnt mind
wow! so great!~~
I think I love to remember this than Trig. Identities.
Nice
Thanks a lot
hay!
I am waiting
KindlySir my problem will be solve
It doesn't work in my calculator.. can someone give an example?
is this the Euler's equation somehow?
This is Euler's Formula, as he states in the open bit of the video. Plug in pi for theta and you get e^(i*pi)=1.
kindlin You dropped a minus sign.
My bad, I was just typing that up real quick. didn't do any proofing lol.
Your identity: cos(z)= (e^iz+e^-iz)/2
My identity:cos(iz)=cosh(z)
Hey, "master of math". Did you forget to mention that z is not a complex number itself? Meaning that your formula proof works ONLY when z is real?
How about cos(x+iy) ? You forgot that you deal with complex numbers? How come?
thank you
thanks
Genius
Kindly solved my problem i am waiting
so we can solve sin(x)=a for any a
Yes but to do that you need the find out the argument(the angle between th number and the x axis) of the number for which you have to use inverse trig functions of real variables -1 to 1 So you cant cheat with it.
I'm not so sure about the first step because exp(z)=exp(x)*(cos(y)+i*sin(y)) by definition.
could you please clarify why we are allowed to leave out the exp(x) ?
cosh(z)=cos(z/i)
=cos(zi/ii)
=cos(zi/-1)
=cos(iz)
i doesn't like to be on the bottom, i like to be on the top !
Define them in terms of the lambert W function and you'll win the 8th millennial prize. It's a secret btw. The first 7 are just to distract you xD
Well played willy wonka, well played.
Sin(tau/4-theta)=cos(theta) just do that
Tau ? You probably meant 2pi...
Wait. So cos(iz)=cosh(z) omg now we can calculate sins of imaginary numbers
Zed
Fun fact: cos i is real
Hey there!
Great fan of yours.
But here lies a slight CONCEPTUAL ERROR in your solution to this!
EULER THEOREM is only valid for REAL inputs / numbers
You have taken (z) as a complex number and hence cannot be applied in the theorem.
Thus making the solution COMPLETELY WRONG!!!!
Please check on it once and reply ASAP
This was a humble notification ( no offences)
☺️
Stop this madness and define it with power series...
Why is it madness? Power series are how you get Euler's formula, aren't they?
There is not one way to "get" Euler's formula or, for that matter, to define sine and cosine. There are multiple equivalent definitions that can be inter-proved with each other. The most intuitive definition is that sine and cosine coordinatize the unit circle with a uniform speed parameterization, but this does not really help in extending to complex numbers - the so-called "analytical extension". Taylor series provide the analytic extension; so it it is better to first develop them from the real number plane definitions of sine and cosine, then use them as definitions of the _complex_ sine and cosine functions. You could also use the differential equations:
df/dz = f, f(0) = 1
does also work on the complex numbers to define the exponential, and
d^2 f/dz^2 = -f, f(0) = 0
does also define the complex sine (and if f(0) = 1, complex cosine. But this I think may require even more complex analysis to prove the complex analogues of the differential equation rules work, than just the basics with analytical extensions via Taylor series. The limit
exp(z) = lim_{n->oo} (1 + z/n)^(1/n)
also well serves to define exp on the complex plane just as much as it does for real numbers, and you can prove that exp(iz) = cos(z) + i sin(z) without even needing an analytical extension for cosine and sine beyond their real-number definitions as just mentioned.
𝐭𝐡𝐚𝐧𝐤𝐬 𝐬𝐨 𝐦𝐮𝐜𝐡 𝐛𝐫𝐨
Thank you