I enjoyed all your Mathematics lectures. Brilliant teacher with excellent explanations of these concepts. I am sure your university is proud to have you as an exceptional Math & Engineering lecturer.
I used complex numbers extensively for my electrical engineering degree. It is interesting to see that they are actually part of a whole section on math called Complex Analysis.
@anonymousalien2099 I was taught it, but it was never presented as a mathematical discipline. It was presented in bits and pieces as was necessary to move through the curriculum.
This is 100% my favorite part of math. I remember spending like three years wondering where That One Big Book got all these integrals on the table of integrals that had pi in them. Then I learned, and it was wonderful.
Incredibly concise, efficient, and well presented. Thank you for sharing! Love watching your lecture series and catching up on some of the math I saw in my undergraduate applied mathematics program, and seeing the true usefulness and beauty of it all a second time around!
Great lecturer! I watched many of the lectures on your channel, and they are always explained with brightness - diving into the deep details while keeping the highlevel intuition and motivation clear. Well done and thanks!
This is nice lecture Sir, I did basic complex numbers up to De moivre' theorem, to solve electrics circuits at diploma level but did not knows that its extended to be called complex Analysis. thanks for this, its helpful, I am a math hobby from electric circuits to mathematics liking, cool stuffs!!
Thank you so much for creating these, your video lectures on dynamical systems and the control boot camp have greatly inspired me and have helped a lot with understanding the content in my classes!! Thanks!!
Nice one, a new topic. Not going to lie but the last mini course got away on me towards the end. Still learned heaps at the start though. Looking forward to see how far this one takes me. Thanks for putting the time into these courses.
Thank you for structured, thorough, compact explanations in your overview lecture. Grateful you defined next topics in series. Real-world application examples always helpful- thanks for explaining applications of topic.
@10.54 - "we can think of the position as the real component of the pendulum, and the velocity as the imaginary component" - this seems fine if the two terms stay I dependent of each other, but how do I understand that in circumstances where the complex number is squared, the i^2 product term stops referring to velocity and now refers to position? Also, doesn't this break dimensional analysis, eg how does m/s units multiplied by m/s units suddenly refer to m?
Hi Steve, this is amazing! I understood the concepts in the abstract, but I see clearly now. You promise a link at around 6:10 on the history of complex numbers that I don't see in the links. Would you mind sharing?
A great teacher and book author as well. I recommend purchase his academic books and consider few of them as official textbooks in their subjects. His youtube channel is an easy explanation-lessons of his books and may consider as good complements materials beside his text-books.
Thanks for another great series. Hope you can fit in Roots of Unity and Quaternions. While they might be considered ‘advanced’ areas, the generalized concepts helped me to understand complex numbers in the same way that the generalized Stokes Thm helps vector calculus make sense.
Z1 and Z2 can give you more of an issue when solving with your said radius and complex numbers. Try using no imaginary axis. Any number Z will allow its position to be neutral.
Hey Steve. I have been loving your channel and been binge watching. Are you thinking of doing some videos of more in depth MPC implementation (preferably with real-time flight trajectory control), and also SMC and the Lyapunov Control Function + Asymptotic stability? Reason for the latter is me wanting to include derivatives in my control input vector (U) which I can't do with LQR or other linearized methods.
@@vladimir10 It's called: "Engineering Math: Crash Course in Complex Analysis". I can find it in the playlists section. This seems to be the url: ua-cam.com/play/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg.html
My man, you will never elucidate the mystery of COMPLEX NUMBERS with ANY of these methods! It is something much much more beautiful, bot for that you have to get to know this: O|O ;)
Takeaways: Complex analysis is the study of complex variables (numbers and functions) Complex variables are written as x + iy where x is real and y is imaginary The mini-lecture series will be about 12 or 13 mini lectures Complex functions and variables come up in differential equations and physics (e.g. pendulum swinging, mass on a spring) Sines and cosines are the real and imaginary parts of a complex exponential function The imaginary number "i" is defined as the square root of negative one and has troubled mathematicians for centuries Negative numbers and imaginary numbers were not considered real at first but have become an established part of mathematics over time Gauss formalized the use of imaginary numbers to write down the solution of a generic polynomial. Complex valued numbers are necessary and sufficient to express the roots of a polynomial. Complex numbers are rich enough to capture the solution of all polynomials with real or complex coefficients. Complex numbers come up frequently in various mathematical areas such as ordinary and partial differential equations, fluid dynamics, quantum mechanics, electromagnetism, etc. Complex numbers are thought of as the real and imaginary components of a complex function. Despite their frequent appearance in the physical world, the concept of imaginary numbers is still confusing and has been so for hundreds of years. The transcript is a lecture on complex numbers in polar coordinates Topics discussed include addition and subtraction, multiplication, and division of complex numbers explains that the real parts of complex numbers add or subtract and imaginary parts add or subtract in the case of addition or subtraction In the case of multiplication, the speaker shows how to multiply two complex numbers and the result is split into real and imaginary parts The speaker mentions that using polar coordinates makes multiplication and division easier to perform The speaker concludes by mentioning that division is performed in a similar manner to the way it is done in high school or middle school math. A complex number can be represented as "r * e^(i * theta)" "Theta" is the angle of the complex number and "r" is the radius "e^(i * theta)" is based on Euler's formula, where e^(i * theta) = cos(theta) + i * sin(theta) Complex multiplication is easier when the complex numbers are represented in polar form (using "r * e^(i * theta)") Two complex numbers can be multiplied as: r1 * e^(i * theta1) * r2 * e^(i * theta2) = r1 * r2 * e^(i * (theta1 + theta2)) "z^2" is a complex function where squaring a complex number "z" will result in x^2-y^2+i*2xy Functions like polynomials (z^n), trigonometric (sin(z), cos(z)), exponential (e^z), and logarithmic (Log(z)) can be extended to be functions of complex variable "z" These functions are called analytic functions and the real and imaginary parts are solutions to Laplace's equation Laplace's equation and its solutions play a crucial role in partial differential equations such as electromagnetism, heat equation, wave equation The next 12-13 lectures will cover topics such as calculus of complex variables, derivatives and integrals, Euler's formula for Taylor series, and more
I think you can get rid of this "imaginary" notion of complex numbers, by showing that the complex numbers form a 2D real vector space R2. This follows from the definition of addition/subtraction of complex numbers. The proof is very simple. This is the justification for depicting the complex numbers in a cartesian coordinate system, i.e. the "complex plane".
A certain form of multiplication is defined on this 2D-vectorspace. The term i^2=-1 is just a short hand notation for multiplying the 2D-vectors with each other on this vector space. It doesn't mean that some strange inexplicable imaginary number exists that is negative, if you multiply it with itself.
Interesting clip but you start from the first seconds with a false premise that is so rampant among academics and it is very sad and disappointing. Basically, there exist nothing called i=sqrt(-1) That is a non-existent definition and nowhere in math history any reliable source has ever defined it. The true and unique definition is i^2 = -1 and from that no one can possibly deduce i= sqrt(-1). To make this clear once for all, let's (for the sake of argument) assume that i=sqrt(-1) exist (again, it doesn't but we pretend it does!). Then we will have: -1 = i^2 = i*i = sqrt(-1) * sqrt(-1) = sqrt( -1 * -1) = sqrt (+1) = +1 which wrongly implies -1 = +1 Besides, try the Euler's equation itself: e^(ix) = Cos(x) + i Sin(x) and try to substitute i with sqrt(-1) and see if you get the same result which of course you don't because i is the complex number and can't be substituted by any other definition. Another obvious example would be e^(i * Pi) = -1. Try to substitute i with sqrt(-1) in that and see if e^(sqrt(-1) * Pi) will be equal to -1 which you won't be able to show. I hope that you and all others are now clear about this very common and unfortunate mistake and actually mis-definition of the complex number "i" and won't use that false definition from now on. Good Luck!
me: just skipping 5 minutes video 🤯🤯🤯: z^4 + (3 + 2i)z^3 - (2 + 5i)z^2 - (1 + 4i)z + 6 = 0 where z is a complex number of the form z = a + bi, where a and b are real numbers, and i is the imaginary unit. To solve this equation, we can apply complex number algebra and factoring techniques. By factoring out z, we can rewrite the equation as: z(z^3 + (3 + 2i)z^2 - (2 + 5i)z - (1 + 4i)) + 6 = 0 Now, we can focus on solving the cubic equation inside the parentheses. To find the roots of the cubic equation, we can use methods like synthetic division, numerical approximation, or software tools. Once we find the roots of the cubic equation, we can substitute them back into the original equation to find the values of z that satisfy the equation.
I enjoyed all your Mathematics lectures. Brilliant teacher with excellent explanations of these concepts. I am sure your university is proud to have you as an exceptional Math & Engineering lecturer.
I've learnt my engineering mathematics poorly in my student time. Thanks very much for sharing and can't wait to learn from this new series!
I used complex numbers extensively for my electrical engineering degree. It is interesting to see that they are actually part of a whole section on math called Complex Analysis.
@anonymousalien2099 I was taught it, but it was never presented as a mathematical discipline. It was presented in bits and pieces as was necessary to move through the curriculum.
@anonymousalien2099 nothing is taught as a subejct or field but in bits and pieces during engineering maths... You just needs these tools ..
it goes bloody deep let me tell you
This is 100% my favorite part of math. I remember spending like three years wondering where That One Big Book got all these integrals on the table of integrals that had pi in them. Then I learned, and it was wonderful.
Incredibly concise, efficient, and well presented. Thank you for sharing! Love watching your lecture series and catching up on some of the math I saw in my undergraduate applied mathematics program, and seeing the true usefulness and beauty of it all a second time around!
Great lecturer! I watched many of the lectures on your channel, and they are always explained with brightness - diving into the deep details while keeping the highlevel intuition and motivation clear. Well done and thanks!
WOW! I learned a lot in 30 min.! Whets my appetite for complex analysis! Thanx so much! 😊
This is nice lecture Sir, I did basic complex numbers up to De moivre' theorem, to solve electrics circuits at diploma level but did not knows that its extended to be called complex Analysis. thanks for this, its helpful, I am a math hobby from electric circuits to mathematics liking, cool stuffs!!
Really nice series, just jumped from the potential flow to here for more details.
Thank you so much for creating these, your video lectures on dynamical systems and the control boot camp have greatly inspired me and have helped a lot with understanding the content in my classes!! Thanks!!
Just found this. Chad playlist
@eigensteve is a gigachad
Nice one, a new topic. Not going to lie but the last mini course got away on me towards the end. Still learned heaps at the start though.
Looking forward to see how far this one takes me. Thanks for putting the time into these courses.
nice overview of complex numbers. complex numbers the key to digital comms and iq modulators
Thank you for structured, thorough, compact explanations in your overview lecture. Grateful you defined next topics in series. Real-world application examples always helpful- thanks for explaining applications of topic.
just in time , I'm going to be taking a course in complex variables starting from monday!
Great video!
Really nice!
@10.54 - "we can think of the position as the real component of the pendulum, and the velocity as the imaginary component" - this seems fine if the two terms stay I dependent of each other, but how do I understand that in circumstances where the complex number is squared, the i^2 product term stops referring to velocity and now refers to position? Also, doesn't this break dimensional analysis, eg how does m/s units multiplied by m/s units suddenly refer to m?
Nice video. Simple straightforward and direct to the point. Thank you
Very Nice Lecture
I liked it...👍👍
Hi Steve, this is amazing! I understood the concepts in the abstract, but I see clearly now. You promise a link at around 6:10 on the history of complex numbers that I don't see in the links. Would you mind sharing?
I wish you can prepare more math courses for the undergrad level, your explanation is great
Thank you a lot! ♥♥♥
Awsome thank you teacher also your tone of voice is brilliant 😃
Great lectures on complex analysis. 😤Thank you.
Clear and lucid presentation, thank you!
A great teacher and book author as well. I recommend purchase his academic books and consider few of them as official textbooks in their subjects. His youtube channel is an easy explanation-lessons of his books and may consider as good complements materials beside his text-books.
thank you, Steve
dude... this video is amazing
If I had a million dollars I would spend all of it to have Dr Steve make videos on maths, control theory and adv machine learning and signal analysis.
This man is a literal god
Could you try eliminating the pendulum and after reintroduce said pendulum in all directions trying at different velocity
wow your explination is amaizng we are looking for the next chapters :)
Unreal .. I mean really good! .. thanks!
Thanks for another great series.
Hope you can fit in Roots of Unity and Quaternions. While they might be considered ‘advanced’ areas, the generalized concepts helped me to understand complex numbers in the same way that the generalized Stokes Thm helps vector calculus make sense.
Z1 and Z2 can give you more of an issue when solving with your said radius and complex numbers. Try using no imaginary axis. Any number Z will allow its position to be neutral.
you're The Best!
This is really helpful thank you so much
Is it possible to send me pdf of youtube lectures on complex analysis
Hey Steve. I have been loving your channel and been binge watching. Are you thinking of doing some videos of more in depth MPC implementation (preferably with real-time flight trajectory control), and also SMC and the Lyapunov Control Function + Asymptotic stability? Reason for the latter is me wanting to include derivatives in my control input vector (U) which I can't do with LQR or other linearized methods.
28:08 really upgraded my gray matter
Will there be a dedicated playlist for complex analysis? Do you intend to talk about multivariable complex functions as well?
there is one already, all lectures are there
@@jesperheuver5779 Which one? I didn't see in the playlists section
@@vladimir10 It's called: "Engineering Math: Crash Course in Complex Analysis". I can find it in the playlists section. This seems to be the url: ua-cam.com/play/PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg.html
@@jesperheuver5779 oh, thanks for the link, I'll checked that out!
Thank you
The Veritasium video about imaginary numbers: ua-cam.com/video/cUzklzVXJwo/v-deo.html
I was half expecting a rick roll to come my way
Good primer on complex numbers :)
Very beautiful 👌
"1 unavailable video is hidden" on the Complex analysis playlist. Can we get it (back), please? :)
i kept getting distracted. How does that board work...
Based Analysis
I want to know the history of iota...
Why we need it and from where it comes.....
My man, you will never elucidate the mystery of COMPLEX NUMBERS with ANY of these methods! It is something much much more beautiful, bot for that you have to get to know this: O|O ;)
Amazing ---
What is limit of math
awesome
Im here to build up for the inverse laplace. I gotta be able to do this by hand, i hate these damn tables more than anything.
I wonder why proffesors just make easy things like these look so complicated
How is he writing things in the board in inverted form ? anyone?
Mirrored
exercises?
Take a look at part D, chapters 13-18, of "Advanced Engineering Mathematics" by Erwin Kreyszig
They also come up in alternating current electrical networks analysis. That is how I know imaginary numbers exist.😅
The insane part about this aside from the math: he’s writing everything backwards
The video is mirrored before uploading
Takeaways:
Complex analysis is the study of complex variables (numbers and functions)
Complex variables are written as x + iy where x is real and y is imaginary
The mini-lecture series will be about 12 or 13 mini lectures
Complex functions and variables come up in differential equations and physics (e.g. pendulum swinging, mass on a spring)
Sines and cosines are the real and imaginary parts of a complex exponential function
The imaginary number "i" is defined as the square root of negative one and has troubled mathematicians for centuries
Negative numbers and imaginary numbers were not considered real at first but have become an established part of mathematics over time
Gauss formalized the use of imaginary numbers to write down the solution of a generic polynomial.
Complex valued numbers are necessary and sufficient to express the roots of a polynomial.
Complex numbers are rich enough to capture the solution of all polynomials with real or complex coefficients.
Complex numbers come up frequently in various mathematical areas such as ordinary and partial differential equations, fluid dynamics, quantum mechanics, electromagnetism, etc.
Complex numbers are thought of as the real and imaginary components of a complex function.
Despite their frequent appearance in the physical world, the concept of imaginary numbers is still confusing and has been so for hundreds of years.
The transcript is a lecture on complex numbers in polar coordinates
Topics discussed include addition and subtraction, multiplication, and division of complex numbers
explains that the real parts of complex numbers add or subtract and imaginary parts add or subtract in the case of addition or subtraction
In the case of multiplication, the speaker shows how to multiply two complex numbers and the result is split into real and imaginary parts
The speaker mentions that using polar coordinates makes multiplication and division easier to perform
The speaker concludes by mentioning that division is performed in a similar manner to the way it is done in high school or middle school math.
A complex number can be represented as "r * e^(i * theta)"
"Theta" is the angle of the complex number and "r" is the radius
"e^(i * theta)" is based on Euler's formula, where e^(i * theta) = cos(theta) + i * sin(theta)
Complex multiplication is easier when the complex numbers are represented in polar form (using "r * e^(i * theta)")
Two complex numbers can be multiplied as: r1 * e^(i * theta1) * r2 * e^(i * theta2) = r1 * r2 * e^(i * (theta1 + theta2))
"z^2" is a complex function where squaring a complex number "z" will result in x^2-y^2+i*2xy
Functions like polynomials (z^n), trigonometric (sin(z), cos(z)), exponential (e^z), and logarithmic (Log(z)) can be extended to be functions of complex variable "z"
These functions are called analytic functions and the real and imaginary parts are solutions to Laplace's equation
Laplace's equation and its solutions play a crucial role in partial differential equations such as electromagnetism, heat equation, wave equation
The next 12-13 lectures will cover topics such as calculus of complex variables, derivatives and integrals, Euler's formula for Taylor series, and more
👏🏻👏🏻👏🏻👏🏻🧛🏻♂️❤️
Learning maths from thawne (reverse flash)😂
Hello, it's awesome but please to add this videos to a new playlist.
the parenthetical jokes omg . power series.
I think you can get rid of this "imaginary" notion of complex numbers, by showing that the complex numbers form a 2D real vector space R2. This follows from the definition of addition/subtraction of complex numbers. The proof is very simple. This is the justification for depicting the complex numbers in a cartesian coordinate system, i.e. the "complex plane".
A certain form of multiplication is defined on this 2D-vectorspace. The term i^2=-1 is just a short hand notation for multiplying the 2D-vectors with each other on this vector space. It doesn't mean that some strange inexplicable imaginary number exists that is negative, if you multiply it with itself.
Interesting clip but you start from the first seconds with a false premise that is so rampant among academics and it is very sad and disappointing. Basically, there exist nothing called i=sqrt(-1)
That is a non-existent definition and nowhere in math history any reliable source has ever defined it.
The true and unique definition is i^2 = -1 and from that no one can possibly deduce i= sqrt(-1).
To make this clear once for all, let's (for the sake of argument) assume that i=sqrt(-1) exist (again, it doesn't but we pretend it does!). Then we will have:
-1 = i^2 = i*i = sqrt(-1) * sqrt(-1) = sqrt( -1 * -1) = sqrt (+1) = +1
which wrongly implies -1 = +1
Besides, try the Euler's equation itself: e^(ix) = Cos(x) + i Sin(x) and try to substitute i with sqrt(-1) and see if you get the same result which of course you don't because i is the complex number and can't be substituted by any other definition.
Another obvious example would be e^(i * Pi) = -1. Try to substitute i with sqrt(-1) in that and see if
e^(sqrt(-1) * Pi) will be equal to -1 which you won't be able to show.
I hope that you and all others are now clear about this very common and unfortunate mistake and actually mis-definition of the complex number "i" and won't use that false definition from now on.
Good Luck!
me: just skipping 5 minutes
video 🤯🤯🤯: z^4 + (3 + 2i)z^3 - (2 + 5i)z^2 - (1 + 4i)z + 6 = 0
where z is a complex number of the form z = a + bi, where a and b are real numbers, and i is the imaginary unit.
To solve this equation, we can apply complex number algebra and factoring techniques. By factoring out z, we can rewrite the equation as:
z(z^3 + (3 + 2i)z^2 - (2 + 5i)z - (1 + 4i)) + 6 = 0
Now, we can focus on solving the cubic equation inside the parentheses. To find the roots of the cubic equation, we can use methods like synthetic division, numerical approximation, or software tools.
Once we find the roots of the cubic equation, we can substitute them back into the original equation to find the values of z that satisfy the equation.
Again No Matlab,, Sorry Professor, this is not good thing
Big thanks @eigensteve
Hello, it's awesome but please to add this videos to a new playlist.