Complex Connections between Symmetry and Singularity Analysis

Round 1

Reviewer 1 Report

In this manuscript, the topics of Symmetry analysis, Painlevé Analysis, and representations of functions are reviewed. The work of Chaudhry and Qadir and the later findings of Jamshaid and Qadir are covered. Their research discovered new identities for special functions, such as gamma and Riemann zeta. They achieved this by using a singular representation of these functions and improving upon older formulations, making the process simpler and easier to understand. That is a significant contribution to the field, and the manuscript provides an invaluable resource for those studying these topics.

Author Response

The reviewer recommends it, so no action or response is needed.

Reviewer 2 Report

Report on

“Complex Connections Between Symmetry and Singularity Analysis”

by

Asghar Qadir

In this paper, the author mainly summarizes the relevant salient points of singularity analysis, presents three symmetry analysis and the complex methods, represents some special functions, and explores the common aspects of all three applications. After reviewing carefully, I recommend this manuscript after minor revision.

Questions and Suggestions

1.      In Abstract, you just introduced others’ work, could you explain what your main jobs were for convenience of reading?

2.     In two distinct cases of page 4, why is there no q<p?

3.     In page 4, how are c=k and 2d-c=k determined?

4.      In line 3 of Section 4 (Page 9), there is a small printing error of ‘totaly’.

5.      The manuscript involves a lot of others’ jobs. Could you briefly describe the innovative points of your work?

Null

Author Response

Reviewer #2: The reviewer gives five points, to which I respond pointwise.

1. The reviewer asks for the new input of this paper to be given in the “Abstract”. It seems to me that this is because of the very long explanation in the Abstract, whereas the reviewer is looking for a brief, terse statement of what the paper is about. I have modified the Abstract to: “In this paper it is noted that three apparently disparate areas of Mathematics: Singularity Analysis; Complex Symmetry Analysis; and the Distributional Representation of Special Functions; have a basic commonality in the underlying methods used. The insights obtained from the first of these provides a much-needed explanation for the effectiveness of the latter two. The consequent explanations are provided in the form of two theorems and their corollaries.”
2. The reviewer asks why q < p does not arise. In the Frobenius method the singular behaviour comes from the independent variable, x, and so the leading term in it will also “lead” the singular behaviour of the dependent variable, y. Hence the exponent of y, namely q, cannot be less than that of x, namely p; as it would lead the independent variable otherwise. This applies equally for the Painlevé singularity analysis. This should be apparent in the discussion at the end of p.3. I feel that this explanation is not needed there.
3. The reviewer asked how “c=k and 2d-c=k [are] determined”. I had written “One might use the next terms in the Laurent series for …”, which might would have caused confusion. I have changed the wording to, “Using the Laurent series to cancel the next to leading terms for x and y, cτ^p+1 and dτ^q+1, in case (i) …”.
4. The editorial correction pointed out by the Reviewer has been inserted.
5. This point is similar to the first one. The long Abstract seems to have confused the reviewer. As the new Abstract says, it is the insight and explanation that are new. Because there are three fields, and one may expect the reader to be familiar with one, or possibly two, it would be unreasonable to expect it for all three, it is necessary to review all three. The insights are given after the review in the last section, which served as a Conclusion. I have now split it making a separate Conclusion to provide greater clarity that the reviewer asked for.

Reviewer 3 Report

Fresh approach to the basic issue of how complex analysis influences the status of symmetry content of differential equations. Interesting format and engaging presentation that will serve well a wide range of readers.

Author Response

The reviewer recommends it, so no action or response is needed.

Reviewer 4 Report

Descriptions of the field, ie, the history, is well presented and detailed.

The authors need to present some examples which will be useful to readers. It is all very well that the theory and existing results are well presented but the reader could be left `hanging' and asking `where does the paper lead to or what is the significance of it all?'. This has to be taken care of.

Author Response

See the attachment for responses to Reviewer #4.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The author has made corresponding modifications according to the review comments, and it is recommended to accept and publish this version.

Reviewer 4 Report

None