I've gotten tantalizingly close but haven't been able to crack it. What is the closed form for this hypergeometric function, if there is one? *Also, this is probably a very stupid question, but on both Wolfram Mathworld and Wikipedia they claim $_2F_1\left(\frac12,\frac12;\ 1;\ k^2\right)=\operatornameK(k)$ which means $(1)$ should be equal to $\operatornameK(1/3)$ but Wolfram Alpha and Mathematica both seem to say $_2F_1\left(\frac12,\frac12;\ 1;\ k\right)=\operatornameK(k)$
It follows \beginalign*_2F_1\left(\frac12,\frac12;1;\frac19 \right)&=\sum_n=0^\infty\frac\left(\frac12\right)^\overlinen\left(\frac12\right)^\overlinen1^\overlinenn!\left(\frac19\right)^n\\&=\sum_n=0^\infty\frac(2n)!2^2nn!\cdot\frac(2n)!2^2nn!\cdot\frac1n!^2\left(\frac19\right)^n\\&=\sum_n=0^\infty\binom2nn^2\frac1144^n\tag1\endalign* Wolfram alpha evaluates the series with the square of the central binomial coefficient (1) as \beginalign*\sum_n=0^\infty\binom2nn^2\frac1144^n=\frac2\piK\left(\frac19\right)\tag2\endalign* with $K$ the complete elliptic integral of the first kind with parameter $k^2=\frac19$.
Wolfram Alpha Crack
Download File: https://slocinoutteu.blogspot.com/?file=2vzq3P
I have been working on this problem for a while now and am unable to crack it. I understand that it is a first order homogenous differential equation. However, doing a substitution such as: $$ u = \fracyx $$Did not help as I would have to solve the following integral:
The main program is then elaborated with a bit of logic to (1) check whether the input directly addresses either AI, and (2) if not, give WolframAlpha a first crack at it, falling back on GPT if the WolframAlpha API returns an error.
Here's the complete program, in case you're curious or want to try it yourself. import "json"import "stringUtil"import "listUtil"import "dateTime"wolframAppId = "V4Q9R5-UEKX2X6XAV"urlSafeChars = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + "abcdefghijklmnopqrstuvwxyz0123456789-_."urlEncode = function(s) bytes = new RawData bytes.resize s.len * 3 len = bytes.setUtf8(0, s) result = [] for i in range(0, len-1) b = bytes.byte(i) c = char(b) if urlSafeChars.contains(c) then result.push c else result.push "%" + hex2(b) end if end for return result.join("")end functionwolframCheckQuery = function(query) url = " " + "?appid=" + wolframAppId + "&mode=Default" + "&output=json" + "&i=" + urlEncode(query) globals.rawResult = http.get(url) globals.result = json.parse(rawResult) if result.hasIndex("error") and result.error then print "Error: " + result.msg return false end if return result.query.acceptedend function// tokens Alpha uses to maintain conversational context:conversationID = ""conversationS = ""wolframResponse = function(query) url = " " + "?appid=" + wolframAppId + "&i=" + urlEncode(query) if conversationID then url = url + "&conversationid=" + conversationID end if if conversationS then url = url + "&s=" + conversationS end if globals.rawResult = http.get(url) globals.result = json.parse(rawResult) if result.hasIndex("error") and result.error then return "Error: " + result.error end if globals.conversationID = result.conversationID if result.hasIndex("s") then globals.conversationS = result.s end if return result.resultend functioncontext = []context.push "Assistant is a large language model capable of "context.push "helping the user in many ways. Also present is "context.push "WolframAlpha, which is very good at math and "context.push "numerical facts."context.push "Knowledge cutoff: 2022-09"context.push "Current date: " + dateTime.now("yyyy-MM-dd")apiKey = function if outer.hasIndex("_apiKey") then return _apiKey data = file.readLines("/usr/API-key.txt") if data == null then print "API-key.txt file not found." exit end if outer._apiKey = data[0] return _apiKeyend functiongptResponse = function(prompt, temperature=0.5) url = " " headers = headers["Content-Type"] = "application/json" headers["Authorization"] = "Bearer " + apiKey data = data.model = "text-davinci-003" data.prompt = prompt data.temperature = temperature data.max_tokens = 2048 globals.rawResult = http.post(url, json.toJSON(data), headers) globals.result = json.parse(rawResult) if result == null or not result.hasIndex("choices") then return rawResult end if return result.choices[0].text.trimend functionsplitAddress = function(s, possibleNames) slower = s.lower for name in possibleNames if slower.startsWith(name) then result = [name] s = s[name.len:] if s[0] == "," then s = s[1:] s = s.trim result.push s return result end if end for return falseend functionwolframNames = ["wolframalpha", "wolfram alpha", "wolfram", "alpha", "wa"]gptNames = ["gpt", "chatgpt", "openai", "assistant"]clearprint "AI Wonder Twins, unite!"_printMark "(Enter `quit` to exit.)"lastInput = ""while true inp = input(">") if inp.lower == "quit" or inp.lower == "exit" then break // figure out who should take this query (and strip off any direct address) resp = "" wolframQuery = splitAddress(inp, wolframNames) if wolframQuery then inp = wolframQuery[1] responder = "WolframAlpha" if inp == "?" then inp = lastInput end if gptQuery = splitAddress(inp, gptNames) if gptQuery then inp = gptQuery[1] responder = "Assistant" if inp == "?" then inp = lastInput end if if not wolframQuery and not gptQuery then resp = wolframResponse(inp) if resp.startsWith("Error:") then resp = "" responder = "Assistant" else responder = "WolframAlpha" end if end if lastInput = inp context.push "User: " + inp oldColor = text.color if responder == "WolframAlpha" then if not resp then resp = wolframResponse(inp) context.push responder + ": " + resp text.color = "#66CC66" else context.push responder + ": " resp = gptResponse(context.join(char(13))) context[-1] = context[-1] + resp text.color = color.aqua responder = "GPT" end if for line in (responder + ": " + resp).wrap print line end for text.color = oldColorend while Enter fullscreen mode Exit fullscreen mode
However, the question of how to fix the situation at its core in a just and democratic way is a tough nut to crack. At least we can try to influence policies and provide people with information and alternatives to preserve the rights that are at the core of the constitutions of most of our governments.
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