-D4GDdxJrpg.txt

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okay today I'm speaking about the first

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okay today I'm speaking about the first
 

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okay today I'm speaking about the first
of the three great partial differential

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of the three great partial differential
 

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of the three great partial differential
equation partial differential equations

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equation partial differential equations
 

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equation partial differential equations
so this one is called Laplace's equation

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so this one is called Laplace's equation
 

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so this one is called Laplace's equation
named after Laplace and you see partial

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named after Laplace and you see partial
 

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named after Laplace and you see partial
derivatives so we have I don't have time

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derivatives so we have I don't have time
 

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derivatives so we have I don't have time
this equation is in steady state I have

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this equation is in steady state I have
 

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this equation is in steady state I have
x and y I'm in the XY plane and I have

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x and y I'm in the XY plane and I have
 

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x and y I'm in the XY plane and I have
second derivatives in X and in Y so I'm

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second derivatives in X and in Y so I'm
 

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second derivatives in X and in Y so I'm
looking for solutions to that equation

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looking for solutions to that equation
 

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looking for solutions to that equation
and of course I'm given some boundary

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and of course I'm given some boundary
 

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and of course I'm given some boundary
values

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values
 

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values
so time is not here the boundary values

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so time is not here the boundary values
 

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so time is not here the boundary values
the boundary is in the XY plane maybe a

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the boundary is in the XY plane maybe a
 

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the boundary is in the XY plane maybe a
circle think about a circle in the XY

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circle think about a circle in the XY
 

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circle think about a circle in the XY
plane and on the circle I know the

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plane and on the circle I know the
 

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plane and on the circle I know the
solution U

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solution U
 

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solution U
so the boundary values around the circle

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so the boundary values around the circle
 

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so the boundary values around the circle
or give it and I have to find the

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or give it and I have to find the
 

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or give it and I have to find the
temperature you inside the circle so I

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temperature you inside the circle so I
 

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temperature you inside the circle so I
know the temperature on the boundary I

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know the temperature on the boundary I
 

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know the temperature on the boundary I
let it settle down and I want to know

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let it settle down and I want to know
 

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let it settle down and I want to know
the temperature inside and the beauty is

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the temperature inside and the beauty is
 

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the temperature inside and the beauty is
it solves that

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it solves that
 

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it solves that
basic partial differential equation so

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basic partial differential equation so
 

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basic partial differential equation so
let's find some solutions they might not

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let's find some solutions they might not
 

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let's find some solutions they might not
match the boundary values but we can use

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match the boundary values but we can use
 

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match the boundary values but we can use
them so u equal constant certainly

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them so u equal constant certainly
 

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them so u equal constant certainly
solves the equation u equal x the second

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solves the equation u equal x the second
 

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solves the equation u equal x the second
derivatives will be 0 u equal Y here's a

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derivatives will be 0 u equal Y here's a
 

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derivatives will be 0 u equal Y here's a
better one x squared minus y squared so

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better one x squared minus y squared so
 

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better one x squared minus y squared so
the second derivative in the x-direction

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the second derivative in the x-direction
 

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the second derivative in the x-direction
is 2 the second derivative in the

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is 2 the second derivative in the
 

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is 2 the second derivative in the
y-direction is minus 2 so I have 2 minus

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y-direction is minus 2 so I have 2 minus
 

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y-direction is minus 2 so I have 2 minus
2 it solves the equation or this one the

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2 it solves the equation or this one the
 

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2 it solves the equation or this one the
second derivative in X is 0 second

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second derivative in X is 0 second
 

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second derivative in X is 0 second
derivative in Y is 0 those are simple

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derivative in Y is 0 those are simple
 

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derivative in Y is 0 those are simple
solutions but those are only a few

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solutions but those are only a few
 

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solutions but those are only a few
solutions and we need an infinite

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solutions and we need an infinite
 

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solutions and we need an infinite
sequence because we're going to match

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sequence because we're going to match
 

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sequence because we're going to match
initial

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initial
 

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initial
conditions ok so is there a path pattern

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conditions ok so is there a path pattern
 

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conditions ok so is there a path pattern
here so this is degree zero constant

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here so this is degree zero constant
 

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here so this is degree zero constant
these are degree one linear these are

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these are degree one linear these are
 

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these are degree one linear these are
degree two quadratic so I hope for two

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degree two quadratic so I hope for two
 

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degree two quadratic so I hope for two
cubic ones and then I hope for two

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cubic ones and then I hope for two
 

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cubic ones and then I hope for two
fourth degree ones and that's the

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fourth degree ones and that's the
 

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fourth degree ones and that's the
pattern that's the pattern let me find

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pattern that's the pattern let me find
 

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pattern that's the pattern let me find
let me spot the

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let me spot the
 

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let me spot the
the cubic ones X cube if I start with X

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the cubic ones X cube if I start with X
 

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the cubic ones X cube if I start with X
cube of course the second X derivative

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cube of course the second X derivative
 

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cube of course the second X derivative
is probably 6 X

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is probably 6 X
 

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is probably 6 X
so I need the second Y derivative to be

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so I need the second Y derivative to be
 

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so I need the second Y derivative to be
minus 6x and I think minus 3x y squared

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minus 6x and I think minus 3x y squared
 

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minus 6x and I think minus 3x y squared
does it mine the second derivative of in

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does it mine the second derivative of in
 

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does it mine the second derivative of in
Y is 2 times the minus 3x is minus 6x

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Y is 2 times the minus 3x is minus 6x
 

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Y is 2 times the minus 3x is minus 6x
cancels the 6 X from that's from the

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cancels the 6 X from that's from the
 

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cancels the 6 X from that's from the
second derivative there and it works so

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second derivative there and it works so
 

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second derivative there and it works so
that fits the pattern but what is the

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that fits the pattern but what is the
 

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that fits the pattern but what is the
pattern ok here it is it's fantastic

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pattern ok here it is it's fantastic
 

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pattern ok here it is it's fantastic
it's I

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it's I
 

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it's I
get I get these

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get I get these
 

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get I get these
crazy polynomials from taking X plus iy

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crazy polynomials from taking X plus iy
 

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crazy polynomials from taking X plus iy
to the different powers here to the

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to the different powers here to the
 

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to the different powers here to the
first power if n is 1

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first power if n is 1
 

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first power if n is 1
and I just have X plus iy and I take the

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and I just have X plus iy and I take the
 

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and I just have X plus iy and I take the
real part that's X so I'll take a real

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real part that's X so I'll take a real
 

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real part that's X so I'll take a real
part of this

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part of this
 

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part of this
the real part of this when n is 1 the

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the real part of this when n is 1 the
 

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the real part of this when n is 1 the
real part is X

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real part is X
 

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real part is X
what about when n is 2 can you can you

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what about when n is 2 can you can you
 

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what about when n is 2 can you can you
square that in your head so we have x

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square that in your head so we have x
 

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square that in your head so we have x
squared and we have I squared Y squared

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squared and we have I squared Y squared
 

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squared and we have I squared Y squared
I squared be minus 1 so I have x squared

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I squared be minus 1 so I have x squared
 

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I squared be minus 1 so I have x squared
and I have minus y spread look the real

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and I have minus y spread look the real
 

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and I have minus y spread look the real
part of this when n is 2 the real part

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part of this when n is 2 the real part
 

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part of this when n is 2 the real part
of X plus I Y squared the real part is x

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of X plus I Y squared the real part is x
 

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of X plus I Y squared the real part is x
squared minus y squared and the

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squared minus y squared and the
 

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squared minus y squared and the
imaginary part

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imaginary part
 

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imaginary part
was the 2i X Y so the imaginary part

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was the 2i X Y so the imaginary part
 

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was the 2i X Y so the imaginary part
that multiplies I is the 2xy this is our

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that multiplies I is the 2xy this is our
 

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that multiplies I is the 2xy this is our
pattern when n is 2 and when n is 3 I

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pattern when n is 2 and when n is 3 I
 

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pattern when n is 2 and when n is 3 I
take X plus I Y cubed and that begins

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take X plus I Y cubed and that begins
 

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take X plus I Y cubed and that begins
with X cube like that and then I think

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with X cube like that and then I think
 

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with X cube like that and then I think
that the other real part would be a

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that the other real part would be a
 

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that the other real part would be a
minus 3 XY squared I think you should

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minus 3 XY squared I think you should
 

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minus 3 XY squared I think you should
check that and then there will be an

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check that and then there will be an
 

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check that and then there will be an
imaginary part well I think I could

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imaginary part well I think I could
 

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imaginary part well I think I could
figure out the imaginary part as I think

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figure out the imaginary part as I think
 

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figure out the imaginary part as I think
maybe something like minus is something

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maybe something like minus is something
 

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maybe something like minus is something
like

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like
 

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like
-

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yeah maybe maybe it's

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yeah maybe maybe it's
 

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yeah maybe maybe it's
3y x squared minus y cube something like

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3y x squared minus y cube something like
 

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3y x squared minus y cube something like
that

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that
 

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that
that would be the real part and that

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that would be the real part and that
 

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that would be the real part and that
would be the imaginary part when n is 3

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would be the imaginary part when n is 3
 

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would be the imaginary part when n is 3
and wonderfully wonderfully it works for

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and wonderfully wonderfully it works for
 

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and wonderfully wonderfully it works for
all

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all
 

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all
powers

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powers
 

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powers
exponents n

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exponents n
 

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exponents n
so I have now a sort of pretty big

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so I have now a sort of pretty big
 

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so I have now a sort of pretty big
family of solutions a list a double list

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family of solutions a list a double list
 

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family of solutions a list a double list
really the real parts and the imaginary

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really the real parts and the imaginary
 

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really the real parts and the imaginary
parts for every N so I can use those

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parts for every N so I can use those
 

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parts for every N so I can use those
to solve my find the solution U which

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to solve my find the solution U which
 

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to solve my find the solution U which
I'm looking for the the temperature

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I'm looking for the the temperature
 

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I'm looking for the the temperature
inside the circle right now of course I

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inside the circle right now of course I
 

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inside the circle right now of course I
have a linear equation so

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have a linear equation so
 

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have a linear equation so
if I have several solutions I can

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if I have several solutions I can
 

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if I have several solutions I can
combine them and I still have a solution

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combine them and I still have a solution
 

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combine them and I still have a solution
x plus 7y will be a solution plus 11x

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x plus 7y will be a solution plus 11x
 

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x plus 7y will be a solution plus 11x
squared minus y squared no problem Plus

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squared minus y squared no problem Plus
 

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squared minus y squared no problem Plus
56 times 2xy those are all solutions so

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56 times 2xy those are all solutions so
 

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56 times 2xy those are all solutions so
I'm going to find a solution

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I'm going to find a solution
 

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I'm going to find a solution
my final solution you will be a

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my final solution you will be a
 

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my final solution you will be a
combination of this this this this this

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combination of this this this this this
 

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combination of this this this this this
this this and all the others for higher

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this this and all the others for higher
 

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this this and all the others for higher
n that's going to be my solution and I

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n that's going to be my solution and I
 

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n that's going to be my solution and I
will need that infinite family see

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will need that infinite family see
 

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will need that infinite family see
partial differential equations we move

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partial differential equations we move
 

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partial differential equations we move
up to infinite family of solutions

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up to infinite family of solutions
 

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up to infinite family of solutions
instead of just a couple of null

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instead of just a couple of null
 

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instead of just a couple of null
solutions okay so let me take an example

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solutions okay so let me take an example
 

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solutions okay so let me take an example
let me take an example oh

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my we're taking the region to be a

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my we're taking the region to be a
 

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my we're taking the region to be a
circle

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circle
 

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circle
okay

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so in that circle

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so in that circle
 

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so in that circle
I'm looking for the solution U of x and

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I'm looking for the solution U of x and
 

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I'm looking for the solution U of x and
y and actually in a circle it's pretty

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y and actually in a circle it's pretty
 

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y and actually in a circle it's pretty
natural to use polar coordinates instead

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natural to use polar coordinates instead
 

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natural to use polar coordinates instead
of x and y inside a circle that that's

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of x and y inside a circle that that's
 

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of x and y inside a circle that that's
inconvenient in the xy-plane it's

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inconvenient in the xy-plane it's
 

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inconvenient in the xy-plane it's
equation is

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equation is
 

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equation is
involves x equals square root of 1 minus

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involves x equals square root of 1 minus
 

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involves x equals square root of 1 minus
y squared or something

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y squared or something
 

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y squared or something
I'll switch to polar coordinates R and

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I'll switch to polar coordinates R and
 

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I'll switch to polar coordinates R and
theta

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theta
 

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theta
well you might say

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well you might say
 

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well you might say
remember we had these nice family of

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remember we had these nice family of
 

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remember we had these nice family of
solutions

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solutions
 

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solutions
what is it still good in polar

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what is it still good in polar
 

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what is it still good in polar
coordinates well the fact is it's even

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coordinates well the fact is it's even
 

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coordinates well the fact is it's even
better so the solutions you will be the

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better so the solutions you will be the
 

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better so the solutions you will be the
real part and the imaginary part now

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real part and the imaginary part now
 

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real part and the imaginary part now
what is X plus I Y

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what is X plus I Y
 

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what is X plus I Y
in

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in
 

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in
R and theta well we all know X is R cos

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R and theta well we all know X is R cos
 

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R and theta well we all know X is R cos
theta plus

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theta plus
 

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theta plus
I R

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I R
 

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I R
sine theta and

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sine theta and
 

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sine theta and
that's R

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that's R
 

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that's R
times cos theta plus I sine theta the

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times cos theta plus I sine theta the
 

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times cos theta plus I sine theta the
one unforgettable complex

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Euler's formula e to the I theta

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Euler's formula e to the I theta
 

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Euler's formula e to the I theta
ok

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ok
 

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ok
now I need its nth power the nth power

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now I need its nth power the nth power
 

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now I need its nth power the nth power
of this is wonderful the real part and

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of this is wonderful the real part and
 

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of this is wonderful the real part and
imaginary part of the nth power is R to

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imaginary part of the nth power is R to
 

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imaginary part of the nth power is R to
the nth eetu the i N

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the nth eetu the i N
 

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the nth eetu the i N
see that's my X plus iy to the enth much

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see that's my X plus iy to the enth much
 

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see that's my X plus iy to the enth much
nicer in polar coordinates because I can

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nicer in polar coordinates because I can
 

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nicer in polar coordinates because I can
take the real part and the imaginary

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take the real part and the imaginary
 

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take the real part and the imaginary
part right away it's R to the n cos n

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part right away it's R to the n cos n
 

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part right away it's R to the n cos n
theta and

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theta and
 

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theta and
R to the N sine of theta

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R to the N sine of theta
 

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R to the N sine of theta
these are my solutions my long list of

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these are my solutions my long list of
 

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these are my solutions my long list of
solutions to Laplace's equation and it's

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solutions to Laplace's equation and it's
 

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solutions to Laplace's equation and it's
some combination of those some my final

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some combination of those some my final
 

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some combination of those some my final
thing is going to be some combination of

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thing is going to be some combination of
 

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thing is going to be some combination of
those some combination may be

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those some combination may be
 

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those some combination may be
coefficients a sub n sum I

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coefficients a sub n sum I
 

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coefficients a sub n sum I
can use these

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and I can use these so maybe B sub n R

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and I can use these so maybe B sub n R
 

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and I can use these so maybe B sub n R
to the N sine and theta

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to the N sine and theta
 

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to the N sine and theta
you may wonder what I'm doing but what

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you may wonder what I'm doing but what
 

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you may wonder what I'm doing but what
I'm achieving is to find the a big the

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I'm achieving is to find the a big the
 

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I'm achieving is to find the a big the
general solution of Laplace's equation

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general solution of Laplace's equation
 

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general solution of Laplace's equation
instead of two constants that we had for

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instead of two constants that we had for
 

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instead of two constants that we had for
an ordinary differential equation as C 1

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an ordinary differential equation as C 1
 

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an ordinary differential equation as C 1
and a C 2 here I have these guys go from

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and a C 2 here I have these guys go from
 

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and a C 2 here I have these guys go from
up to infinity and goes up to infinity

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up to infinity and goes up to infinity
 

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up to infinity and goes up to infinity
so I have many solutions and any

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so I have many solutions and any
 

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so I have many solutions and any
combination working so that's the

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combination working so that's the
 

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combination working so that's the
general solution that's the general

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general solution that's the general
 

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general solution that's the general
solution and I would have to match that

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solution and I would have to match that
 

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solution and I would have to match that
now here's the final step and not simple

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now here's the final step and not simple
 

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now here's the final step and not simple
not always simple I have to match this

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not always simple I have to match this
 

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not always simple I have to match this
to the boundary conditions that's what

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to the boundary conditions that's what
 

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to the boundary conditions that's what
will tell me the constants of course as

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will tell me the constants of course as
 

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will tell me the constants of course as
usual C 1 and C 2 came from the matching

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usual C 1 and C 2 came from the matching
 

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usual C 1 and C 2 came from the matching
the conditions now I don't have just C 1

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the conditions now I don't have just C 1
 

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the conditions now I don't have just C 1
and C 2 I have this infinite family of

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and C 2 I have this infinite family of
 

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and C 2 I have this infinite family of
A's infinite family of bees and I have a

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A's infinite family of bees and I have a
 

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A's infinite family of bees and I have a
lot more to match because on the

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lot more to match because on the
 

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lot more to match because on the
boundary here I have to match

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boundary here I have to match
 

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boundary here I have to match
u 0 which is given so I might be given

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u 0 which is given so I might be given
 

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u 0 which is given so I might be given
suppose I was given the u0 equal the

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suppose I was given the u0 equal the
 

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suppose I was given the u0 equal the
temperature was

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temperature was
 

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temperature was
equal one on the top half and on the

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equal one on the top half and on the
 

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equal one on the top half and on the
bottom half say the temperature is

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bottom half say the temperature is
 

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bottom half say the temperature is
minus one

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that's a typical problem I have a

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that's a typical problem I have a
 

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that's a typical problem I have a
circular region a

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circular region a
 

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circular region a
the top half is held at one temperature

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the top half is held at one temperature
 

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the top half is held at one temperature
the lower half is held at a different

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the lower half is held at a different
 

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the lower half is held at a different
temperature I reach equilibrium

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temperature I reach equilibrium
 

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temperature I reach equilibrium
everybody knows that along that line

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everybody knows that along that line
 

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everybody knows that along that line
probably the temperature would be zero

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probably the temperature would be zero
 

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probably the temperature would be zero
by symmetry but what's the temperature

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by symmetry but what's the temperature
 

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by symmetry but what's the temperature
they are halfway up knotti not so easy

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they are halfway up knotti not so easy
 

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they are halfway up knotti not so easy
or anywhere in there well the answer is

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or anywhere in there well the answer is
 

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or anywhere in there well the answer is
you in the middle U of R and theta

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you in the middle U of R and theta
 

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you in the middle U of R and theta
inside is given by

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inside is given by
 

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inside is given by
that formula

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that formula
 

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that formula
that formula and again the a ends and

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that formula and again the a ends and
 

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that formula and again the a ends and
the B ends come by matching the getting

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the B ends come by matching the getting
 

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the B ends come by matching the getting
the right answer on the boundary well

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the right answer on the boundary well
 

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the right answer on the boundary well
there's a big theory there how do I

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there's a big theory there how do I
 

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there's a big theory there how do I
match these that's called a Fourier

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match these that's called a Fourier
 

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match these that's called a Fourier
series that's called a Fourier series so

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series that's called a Fourier series so
 

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series that's called a Fourier series so
I'm finding the coefficients for a

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I'm finding the coefficients for a
 

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I'm finding the coefficients for a
Fourier series the A's and B's that

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Fourier series the A's and B's that
 

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Fourier series the A's and B's that
match a function around the boundary

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match a function around the boundary
 

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match a function around the boundary
and I could match any function and

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and I could match any function and
 

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and I could match any function and
Fourier series is another entirely

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Fourier series is another entirely
 

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Fourier series is another entirely
separate video this we've done the job

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separate video this we've done the job
 

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separate video this we've done the job
with Laplace's equation in a circle

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with Laplace's equation in a circle
 

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with Laplace's equation in a circle
we've reduced the problem to a Fourier

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we've reduced the problem to a Fourier
 

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we've reduced the problem to a Fourier
series problem we have found the general

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series problem we have found the general
 

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series problem we have found the general
solution and then to match it to a

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solution and then to match it to a
 

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solution and then to match it to a
specific given boundary value that's a

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specific given boundary value that's a
 

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specific given boundary value that's a
Fourier series problem so I'll have to

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Fourier series problem so I'll have to
 

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Fourier series problem so I'll have to
put that off to the Fourier series video

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put that off to the Fourier series video
 

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put that off to the Fourier series video
thank you