Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

In summary, the conversation discusses the use of the Hardy Z function and the calculation of pedal coordinates for a curve defined by the function. The individual discussing the topic has written an article and provided a maple procedure for calculating the curve. However, there seems to be a lack of interest and understanding in the topic among other individuals.
  • #1
qbar
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TL;DR Summary
Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z(t) is the Hardy Z function; How to calculate the implicit derivative of the curve defined by $$Re(Y(t(u)+is(u)))=0$$?
Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to calculate the derivative of $$t(u)$$ and $$s(u)$$ for which the parametric form is not available. When I tried to apply the chain rule, I ended up with
$$\frac{dy}{dx} = - \frac{\frac{d}{dx} Y(x + iy)}{\frac{d}{dy}(Y(x + iy))} = i$$
which I don't think is correct. I have written an article about it with more information at https://github.com/crowlogic/Y/blob/master/tanhln1plusZsquared.pdf

Here is a composite image generated by the real and imaginary parts of Y from 7004.5-0.5i to 7005.5+0.5i (showing a region of the neighborhood surrounding the first Lehmer pair) generated with https://github.com/crowlogic/Y/blob/master/src/complex_plot.c:

YrealImagCompositeFrom7004.5-0.5I..7005.5+0.5I.png


Here is the imaginary part vanishing alone over the same range, with aspect ratio maintained
YimagFrom7004.5-0.5I..7005.5+0.5I.png

Here is the same range where the real part vanishes
YrealFrom7004.5-0.5I..7005.5+0.5I.png

Does anyone have any ideas on how to differentiate these curves ? Specifically, where the real part of Y vanishes?

Does there exist an expression for the tangent line of $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u))) = 0 \}$$ ?

There is a maple procedure to calculate the path at https://github.com/crowlogic/Y/blob/master/tracecurve.mplActually, I just realized I can use finite differences.. it doesn't appear that they are pedal curves of one another.. or if they are, the pedal point is not the root
 
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  • #2
No replies? you people are too boring and conventional.
 
  • #3
qbar said:
No replies? you people are too boring and conventional.
Well, this has probably several reasons:
  1. It is too long.
  2. You struck everybody with the pictures so people barely read the rest.
  3. What are your questions? Probably somewhere, but who wants to search for them.
  4. The Hardy function is a very specific tool in number theory, a branch which is not very popular.
  5. Especially not among physicists.
  6. You have several media breaks: changing sites only to be able to understand is a bad method.
  7. You used far too many double $ instead of inline formulas.
  8. Are you sure you didn't simply want to promote your paper? Where is it published?
  9. Our rules require a published reference.
  10. It is too long.
The main reason, however, is likely the lack of interest in problems around the Hardy Z function.
 
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Likes StoneTemplePython, etotheipi, jim mcnamara and 2 others
  • #4
Unfortunately, @qbar went nutzoid at the helpful reply by @fresh_42 and that profane rant (now deleted) resulted in qbar's ban. Thread is now closed.
 
Last edited:
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What is "Implicitly Differentiating"?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly expressed in terms of its independent variable. It involves treating the dependent variable as a function of the independent variable and using the chain rule to find the derivative.

What is the "Vanishing Real Part" of the Hyperbolic Tangent?

The vanishing real part of the hyperbolic tangent refers to the property of the hyperbolic tangent function where its real part approaches zero as the input approaches infinity. In other words, the real part of the hyperbolic tangent function becomes smaller and smaller as the input value increases.

What is the "Square of the Hardy Z Function"?

The Hardy Z function is a complex-valued function that is closely related to the Riemann zeta function. The square of the Hardy Z function refers to the result of multiplying the Hardy Z function by itself.

Why is the Vanishing Real Part of the Hyperbolic Tangent of One Plus the Square of the Hardy Z Function Important?

This property is important in complex analysis and number theory because it is closely related to the distribution of prime numbers. It has also been used in the study of the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.

How is Implicit Differentiation Used in this Context?

In this context, implicit differentiation is used to find the derivative of the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function. This allows us to better understand the behavior of this function and its relationship to other important mathematical concepts.

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