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Let f : ]0,oo[ x ]0,oo[ -> |R be the function satisfying:

/\(x>0,y>0) cos(arctan(y/x)) = f(x,y).

Find a formula for f(x,y) that does not involve trig functions.

/\(x>0,y>0) cos(arctan(y/x)) = f(x,y).

Find a formula for f(x,y) that does not involve trig functions.

f(x,y) = x / (x^2 + y^2)

page 95 in OLDTIMER

page 95 in OLDTIMER

lim(x,y)->(0,0) (x^3 * y) / ((x^2 - y^2)^2 + y^2) = ???

lim(x,y)->(0,0) (x^3 * y) / ((x^2 - y^2)^2 + y^2) = 0

page 99 in OLDTIMER

page 99 in OLDTIMER

lim(x,y)->(0,0) (x-sin(x))*y / ((x^2-y^2)^2 + y^2) = ???

lim(x,y)->(0,0) (x-sin(x))*y / ((x^2 - y^2)^2 + y^2) = 0

hint: x-sin(x) = x^3 * u(x), where lim(x->0) u(x) = 1/6.

further hint: |ab/(a^2+b^2)|<=K.

page 100 in OLDTIMER

hint: x-sin(x) = x^3 * u(x), where lim(x->0) u(x) = 1/6.

further hint: |ab/(a^2+b^2)|<=K.

page 100 in OLDTIMER

Let Y : P(X) -> |R* satisfy:

(1) Y(O)=0.

(2) AcBcX => Y(A)<=Y(B).

(3) Y(AuB) <= Y(A)+Y(B).

Let M be a ring in X.

Let F = { EcX : /\a>0 \/A:-M Y(E+A) < a }.

In what sense can set F be said to be closed?

(1) Y(O)=0.

(2) AcBcX => Y(A)<=Y(B).

(3) Y(AuB) <= Y(A)+Y(B).

Let M be a ring in X.

Let F = { EcX : /\a>0 \/A:-M Y(E+A) < a }.

In what sense can set F be said to be closed?

Define d:P(X)xP(X)->|R*, d(A,B)=Y(A+B).

Notice that d is a oo-pseudometric.

Now, if (1) /\n:-|N A[n]:-F and (2) lim(n->oo) d(A[n],A) = 0, then (3) A:-F.

See pages 21,64 in 2nd measure for relevant theorems.

Notice that d is a oo-pseudometric.

Now, if (1) /\n:-|N A[n]:-F and (2) lim(n->oo) d(A[n],A) = 0, then (3) A:-F.

See pages 21,64 in 2nd measure for relevant theorems.

Find the power series expansion of cos^2(z) about 0.

cos^2(z) = +(n=0 to oo) a[n]*z^(2n)

a[0] = 1

a[n] = (-1)^n * 2^(2n-1) / (2n)!

page 101 in OLDTIMER

a[0] = 1

a[n] = (-1)^n * 2^(2n-1) / (2n)!

page 101 in OLDTIMER

Let 0 < R <= oo.

Let { c[n] }(n=0 to oo) be a sequence of complex numbers.

Let f : {z:-|C : |z|<R} -> |C be defined f(z) = +(n=0 to oo) c[n]*z^n.

Prove that f is infinitely diffable and derive the formula for its derivatives.

Let { c[n] }(n=0 to oo) be a sequence of complex numbers.

Let f : {z:-|C : |z|<R} -> |C be defined f(z) = +(n=0 to oo) c[n]*z^n.

Prove that f is infinitely diffable and derive the formula for its derivatives.

/\k:-|N /\|z|<R f|k|(z) = +(n=k to oo) c[n]* n!/(n-k)! * z^(n-k)

page 102 in OLDTIMER

/\k:-|N /\|z|<R f|k|(z) = +(n=0 to oo) c[n+k]* (n+k)!/(n!) * z^n

page 102 in OLDTIMER

/\k:-|N /\|z|<R f|k|(z) = +(n=0 to oo) c[n+k]* (n+k)!/(n!) * z^n

Let 0 < R <= oo.

Let { c[n] }(n=0 to oo) be a sequence of complex numbers.

Suppose that /\|z|<R +(n=0 to oo) c[n]*z^n = 0.

What can we conclude from this?

Let { c[n] }(n=0 to oo) be a sequence of complex numbers.

Suppose that /\|z|<R +(n=0 to oo) c[n]*z^n = 0.

What can we conclude from this?

0 = c[0] = c[1] = c[2] = ... = c[n] = c[n+1] = ...

page 103 in OLDTIMER

page 103 in OLDTIMER

Let 0 < R <= oo.

Let a[n],b[n] be sequences of complex numbers, indexed from zero.

Suppose that /\|z|<R +(n=0 to oo) a[n]*z^n = +(n=0 to oo) b[n]*z^n.

In the line above we understand that both series converge.

What can we conclude from this?

Let a[n],b[n] be sequences of complex numbers, indexed from zero.

Suppose that /\|z|<R +(n=0 to oo) a[n]*z^n = +(n=0 to oo) b[n]*z^n.

In the line above we understand that both series converge.

What can we conclude from this?

a[n] = b[n] for all n=0,1,2,3,4,...

page 104 in OLDTIMER

page 104 in OLDTIMER

Prove that /\n:-|N 2^(2n) = +(k=0 to n) Newton(2n+1,2k).

page 106 in OLDTIMER

Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.

See page 104 in OLDTIMER.

Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.

See page 104 in OLDTIMER.

Prove that /\n:-|N 2^(2n) = +(k=0 to n) Newton(2n+1,2k+1).

page 107 in OLDTIMER

Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.

See page 104 in OLDTIMER.

Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.

See page 104 in OLDTIMER.