Contradictory (complex) integral transformation

[ManDay]

The Schwarz-Christoffel mapping (a Riemann-mapping) from the unit disk (z-plane) to a twice-symmtric area (a cross, ζ-plane)

$$ \zeta : \mathbf C \to \mathbf C $$

is given by:

$$\frac{ \mathrm{d}\zeta }{ \mathrm{d} z} = \left( \frac{ ( z^2-b^2 ) ( z^2-\frac 1 {b^2} ) }{ ( z^2-a^2 ) ( z^2-\frac 1 {a^2} ) ( z^2-c^2 ) ( z^2-\frac 1 {c^2} ) } \right)^\frac12 $$

where a, b, and c are points in counter-clockwise order in the first quadrant on the unit circle. If the integration constant is chosen such that

$$\zeta ( z ) = \int_0^z \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y)$$

the cross-shape in the ζ-plane to which the unit circle in the z-plane is mapped has a symmetry about the real and the imaginary axis, i.e. ∀ |z| = 1:

$$ \zeta \left( \frac 1 z \right) = \bar \zeta ( z ) $$

$$ \zeta \left( - \frac 1 z \right) = -\bar \zeta ( z ) $$

This is true and evident from numeric evaluation but it is contradicted by:

$$ \zeta \left( \frac 1 z \right) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) + \int_1^{\frac 1 z }\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

Transformation of the second integral with x := 1/y

$$\begin{aligned}\int_1^{\frac 1 z }\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) &= -\int_1^z \mathrm{d}x \frac 1 {x^2} \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}\left(y = \frac 1 x \right) \\
&= -\int_1^z \mathrm{d}x \frac{ \mathrm{d}\zeta }{ \mathrm{d} x} ( x )
\end{aligned}$$

Where the second line follows from the first by simply inserting dζ/dz and distributing the x² factor on the fraction and pulling out pairwise terms of the forms b and 1/b. So we have

$$ \zeta \left( \frac 1 z \right) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) - \int_1^z\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

but ∫₀¹… is purely real and ∫₁^z… is mixed real and imaginary, so this contradicts the symmetry about the real axis from above because

$$ \zeta ( z ) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) + \int_1^z\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

In fact, the transformation turns out to be wrong. ∫₁^1/z… is indeed the complex conjucate of ∫₁^z…, not the negative, so there must be a problem with the derivation above.

Note: The cross in the ζ-plane is mapped such that 0 maps to the center, 1 maps to the tip of the left arm (i.e. negative real part, 0 imaginary part), -1 maps to the tip of the right arm and i, -i respectively, map to tips of the bottom and top arm. Therefore ∫₀¹… points from the center outwards to the left arm and is purealy real while, say, ∫₁ⁱ… points from the tip of the left arm to the tip of the bottom arm and is mixed real and imaginary.

PS: I'm afraid this is again related to choosing the correct branches on the exponentiation in the integral, so that by the (correct) choice of branches, I get the correct result, that is, the complex conjucate. But this seems messed up: Is there a tool which allows me to treat the transformation consistently without thinking through in detail which branch goes where and when so these be equivalent?

PS II: Even despite that vague suspicion, I can't really see how choosing branches could yield the complex conjugate, because that would only yield a sign for the expression (unless I switch branches inside the interval)

PS III: I'm aware of alternative methods to prove the relation, but that doesn't explain the above contradiction.

 

[Mandelbroth]

I might be able to help you, but...your notation is a little confusing. Do you mean $\displaystyle \zeta(z) = \int_0^z \left. dy \frac{d\zeta}{dy}\right|_y$ or $\displaystyle \zeta(z)=\int_0^z \left. \frac{d\zeta}{dy}\right|_y \, dy$?

 

[micromass]

I might be able to help you, but...your notation is a little confusing. Do you mean $\displaystyle \zeta(z) = \int_0^z \left. dy \frac{d\zeta}{dy}\right|_y$ or $\displaystyle \zeta(z)=\int_0^z \left. \frac{d\zeta}{dy}\right|_y \, dy$?

Physicists tend to put the $dy$ before the function that is integrated.

 

[Mandelbroth]

The Schwarz-Christoffel mapping (a Riemann-mapping) from the unit disk (z-plane) to a twice-symmtric area (a cross, ζ-plane)

$$ \zeta : \mathbf C \to \mathbf C $$

is given by:

$$\frac{ \mathrm{d}\zeta }{ \mathrm{d} z} = \left( \frac{ ( z^2-b^2 ) ( z^2-\frac 1 {b^2} ) }{ ( z^2-a^2 ) ( z^2-\frac 1 {a^2} ) ( z^2-c^2 ) ( z^2-\frac 1 {c^2} ) } \right)^\frac12 $$

[...]

$$\zeta ( z ) = \int_0^z \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y)$$

the cross-shape in the ζ-plane to which the unit circle in the z-plane is mapped has a symmetry about the real and the imaginary axis, i.e. ∀ |z| = 1:

$$ \zeta \left( \frac 1 z \right) = \bar \zeta ( z ) $$

$$ \zeta \left( - \frac 1 z \right) = -\bar \zeta ( z ) $$

[...]

This is true and evident from numeric evaluation but it is contradicted by:

$$ \zeta \left( \frac 1 z \right) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) + \int_1^{\frac 1 z }\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

Here is where I can't follow you. If the modulus of z is equal to 1, then that appears to be correct, because this basically inserts the value for which zeta equals 1/z, and if |z|=1, its complex conjugate is identically it's multiplicative inverse. Thus, I see no apparent contradiction.

Check limits of integration below that point.

 

[ManDay]

Here is where I can't follow you. If the modulus of z is equal to 1, then that appears to be correct, because this basically inserts the value for which zeta equals 1/z, and if |z|=1, its complex conjugate is identically it's multiplicative inverse. Thus, I see no apparent contradiction.

Check limits of integration below that point.

I suspect this is a misunderstanding: Yes, the last line you quoted is indeed obtained by just inserting z* = 1/z into the definition, so to speak, of ζ. The actual contradiction is shown in the following (after where you stopped quoting me) and is concluded where I say

[…] this contradicts the symmetry about the real axis from above because […]

Did this clarify it?

 

[Mandelbroth]

I suspect this is a misunderstanding: Yes, the last line you quoted is indeed obtained by just inserting z* = 1/z into the definition, so to speak, of ζ. The actual contradiction is shown in the following (after where you stopped quoting me) and is concluded where I say

[...]

Did this clarify it?

Kind of. Just so I'm clear, what have you defined y to be? What is it in terms of z?

 

[ManDay]

This is meant in functional notation, not system notation. Two functions f(x) and f(y) have the same form (actually, they are the same function). There are no globally bound variables or any such hocus-pocus that is popular among physicists

$$ \frac { \mathrm{d}\zeta } { \mathrm{d} y } (y) \equiv \left. \frac { \mathrm{d}\zeta } { \mathrm{d} z } \right|_{z=y}$$
(meaning y is just bound by the integral, i.e. a placeholder name)

 

[jackmell]

Transformation of the second integral with x := 1/y

$$\begin{aligned}\int_1^{\frac 1 z }\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) &= -\int_1^z \mathrm{d}x \frac 1 {x^2} \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}\left(y = \frac 1 x \right) \\
&= -\int_1^z \mathrm{d}x \frac{ \mathrm{d}\zeta }{ \mathrm{d} x} ( x )
\end{aligned}$$

Where the second line follows from the first by simply inserting dζ/dz and distributing the x² factor on the fraction and pulling out pairwise terms of the forms b and 1/b. So we have

Your notation is awkward and not easy to follow. Try and come up with something better like doing away entirely with the notation \frac{d\zeta}{dz} and just calling it f(z), then you could write the integrand simply as \frac{1}{x^2} f(\frac{1}{x})dx. Now ain't that easier to read?

Also, I do not agree that:

\frac{d\zeta}{dz}\biggr|_{z=x}=\frac{1}{x^2} \frac{d\zeta}{dz}\biggr|_{z=1/x}

 

[ManDay]

Your notation is awkward and not easy to follow. Try and come up with something better like doing away entirely with the notation \frac{d\zeta}{dz} and just calling it f(z), then you could write the integrand simply as \frac{1}{x^2} f(\frac{1}{x})dx. Now ain't that easier to read?

You seem to conveniently forget that there is a derivative involved. How would that be written with your "easy" notation?
$$\frac{\mathrm{d}f}{\mathrm{d}\left[\frac1x\right]}\left(\frac1x\right)$$

Also, I do not agree that:

\frac{d\zeta}{dz}\biggr|_{z=x}=\frac{1}{x^2} \frac{d\zeta}{dz}\biggr|_{z=1/x}

\begin{aligned}<br /> \frac1{x^2}\frac{\mathrm{d}\zeta}{\mathrm{d}z}\left(z=\frac1x\right) &amp;= \frac1{x^2} \left( \frac{ ( \left(\frac1x\right)^2-b^2 ) ( \left(\frac1x\right)^2-\frac 1 {b^2} ) }{ ( \left(\frac1x\right)^2-a^2 ) ( \left(\frac1x\right)^2-\frac 1 {a^2} ) ( \left(\frac1x\right)^2-c^2 ) ( \left(\frac1x\right)^2-\frac 1 {c^2} ) } \right)^\frac12 \\<br /> &amp;= \frac1{x^2} \left( \frac{ x^4 ( 1-x^2 b^2 ) ( 1-x^2 \frac 1 {b^2} ) }{ ( 1-x^2a^2 ) ( 1-x^2 \frac 1 {a^2} ) ( 1-x^2 c^2 ) ( 1-x^2\frac 1 {c^2} ) } \right)^\frac12 \\<br /> &amp;= \left( \frac{ ( \frac 1 {b^2}-x^2 ) ( b^2-x^2 ) }{ ( \frac 1 {a^2} -x^2 ) ( a^2-x^2 ) ( \frac 1 {c^2} -x^2 ) ( c^2-x^2) } \right)^\frac12 \\<br /> &amp;= \frac{\mathrm{d}\zeta}{\mathrm{d}x}(x)<br /> \end{aligned}<br />

(this is contradicted by a similar calculation where you take the complex vonjugate of the whole term, instead, which yields the correct result)

 

[jackmell]

\begin{aligned}<br /> \frac1{x^2}\frac{\mathrm{d}\zeta}{\mathrm{d}z}\left(z=\frac1x\right) &amp;= \frac1{x^2} \left( \frac{ ( \left(\frac1x\right)^2-b^2 ) ( \left(\frac1x\right)^2-\frac 1 {b^2} ) }{ ( \left(\frac1x\right)^2-a^2 ) ( \left(\frac1x\right)^2-\frac 1 {a^2} ) ( \left(\frac1x\right)^2-c^2 ) ( \left(\frac1x\right)^2-\frac 1 {c^2} ) } \right)^\frac12 \\<br /> &amp;= \frac1{x^2} \left( \frac{ x^4 ( 1-x^2 b^2 ) ( 1-x^2 \frac 1 {b^2} ) }{ ( 1-x^2a^2 ) ( 1-x^2 \frac 1 {a^2} ) ( 1-x^2 c^2 ) ( 1-x^2\frac 1 {c^2} ) } \right)^\frac12 \\<br /> &amp;= \left( \frac{ ( \frac 1 {b^2}-x^2 ) ( b^2-x^2 ) }{ ( \frac 1 {a^2} -x^2 ) ( a^2-x^2 ) ( \frac 1 {c^2} -x^2 ) ( c^2-x^2) } \right)^\frac12 \\<br /> &amp;= \frac{\mathrm{d}\zeta}{\mathrm{d}x}(x)<br /> \end{aligned}<br />

Ok, I see it now. Thanks for clearing that up for me. I'll continue working with it as it's an interesting problem for me. I suspect it's a branching issue you're having but don't see it yet.

 

[jackmell]

Where the second line follows from the first by simply inserting dζ/dz and distributing the x² factor on the fraction and pulling out pairwise terms of the forms b and 1/b. So we have

$$ \zeta \left( \frac 1 z \right) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) - \int_1^z\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

but ∫₀¹… is purely real and ∫₁^z… is mixed real and imaginary, so this contradicts the symmetry about the real axis from above because

$$ \zeta ( z ) = \int_0^1 \mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) + \int_1^z\mathrm{d}y \frac{ \mathrm{d}\zeta }{ \mathrm{d} y}(y) $$

I do not get mixed real and imaginary for the second integral when I check your integrals with z=e^{\pi i/12} and integrate over a path which follows the unit circle: Let's take z=e^{\pi i/12} so that, numerically, \zeta(z)\approx 0.9978+0.2422i and thus \overline{\zeta}(z)\approx 0.9978-0.2422i. Now letting the derivative be u(z):
\int_0^1 u(w)dw\approx 0.9978
\int_1^z u(w)dw=\int_0^{\pi/12} u(e^{it}) i e^{it} dt\approx 0.2422i
and therefore:

\zeta(1/z)\approx 0.9978-0.2422i
which agrees with your derivation.

 

[ManDay]

I do not get mixed real and imaginary for the second integral when I check your integrals with z=e^{\pi i/12} and integrate over a path which follows the unit circle: Let's take z=e^{\pi i/12} so that, numerically, \zeta(z)\approx 0.9978+0.2422i and thus \overline{\zeta}(z)\approx 0.9978-0.2422i. Now letting the derivative be u(z):
\int_0^1 u(w)dw\approx 0.9978
\int_1^z u(w)dw=\int_0^{\pi/12} u(e^{it}) i e^{it} dt\approx 0.2422i
and therefore:

\zeta(1/z)\approx 0.9978-0.2422i
which agrees with your derivation.

I don't know your particular choice for a, b, and c, but while ∫₀¹ lies on the end of the left, horizontal arm, ∫₀^exp(iπ/12) is likely to lie on the same edge, which means they have the same real part.

In general, however, ∫₁^exp(iπr) with r ∈ ℝ, does not point strictly vertically and thus has mixed real- and imaginary parts.

 

[jackmell]

I don't know your particular choice for a, b, and c,

I chose your choices:

a=e^{i/2}, b=e^i, c=e^{3/2 i}

I believe I've determined the cause of your problem, at least to my satisfaction, and if this was a final exam question, this is how I would risk answering it: You have

\int_1^{1/z} u(y)dy

and you propose making the substitution x=1/y. That substitution introduces the function u(1/x)=v(x) and v(x) between the endpoints x=1 and x=z has a principal branch cut which if I just integrated numerically without regards to analytic continuitity, would integrate (discontinuously) over a branch-cut. However if I integrated over an analytically-continuous sheet of v(x), I could get the correct answer. But I'm not sure I can determine which sheet of v(x) to integrate over without knowing the answer beforehand.

Conclusions: The substitution x=1/y is not a recommended technique to evaluate the integral above.

Here's mine: