//It's taken me too long to write the comments for this code and at this point I forgot how most of it works so here's some confusing uncommented JS code window.onload = function(){ alert("For some reason, this program only seems to want to work with numbers from 2-5, with anything higher not having it run") var z = prompt("Enter a number to use within the gamma function"); if(z < 2){ throw new Error("Please only enter a number greater than or equal to 2"); } z -= 1; var numbers = [[[1,"x",z],[1,"e","-x"]]]; var equation = document.getElementById("equation"); var test = document.getElementById("test"); var i = 1; var result = ""; var resultList = []; equation.innerHTML += `Γ(${z+1}) = ∫₀^∞ x^${z} e^-x dx`; equation.innerHTML += "
Using the reverse product rule ( ∫ u'v dx = uv - ∫ uv' dx ), the equation above can be expressed as the following:
"; equation.innerHTML += `${numbers[numbers.length-1][0][1]}^{${numbers[numbers.length-1][0][2]}} ${numbers[numbers.length-1][1][1]}^{${numbers[numbers.length-1][1][2]}} - `; numbers = numbers.concat([[[numbers[numbers.length-1][0][0]*numbers[numbers.length-1][0][2],"x",numbers[numbers.length-1][0][2]-1],[1,"e","-x"]]]); while(numbers[numbers.length-1][0][2] >= 1){ equation.innerHTML += `${numbers[numbers.length-1][0][0]}(${numbers[numbers.length-1][0][1]}^{${numbers[numbers.length-1][0][2]}} ${numbers[numbers.length-1][1][1]}^{${numbers[numbers.length-1][1][2]}} - `; numbers = numbers.concat([[[numbers[numbers.length-1][0][0]-1,"x",numbers[numbers.length-1][0][2]-1],[1,"e","-x"]]]); } equation.innerHTML += `${numbers[numbers.length-1][1][1]}^{${numbers[numbers.length-1][1][2]}}` + ")".repeat(z); equation.innerHTML += "]₀^∞"; numbers[0][1][0] *= numbers[0][0][0]; while(i < numbers.length){ numbers[i][0][0] *= numbers[i-1][0][0]; numbers[i][1][0] *= numbers[i][0][0]; i += 1; } i = 0; while(i < numbers.length){ if(i == numbers.length - 1){ result += `${numbers[i][1][0]}${numbers[i][1][1]}^{${numbers[i][1][2]}}`; break; } else if(i == numbers.length - 2){ result += `${numbers[i][0][0]}${numbers[i][0][1]} ${numbers[i][1][0]}${numbers[i][1][1]}^{${numbers[i][1][2]}}`; } else if(i === 0){ result += `${numbers[i][0][1]}^{${numbers[i][0][2]}} ${numbers[i][1][1]}^{${numbers[i][1][2]}}`; } else{ result += `${numbers[i][0][0]}${numbers[i][0][1]}^{${numbers[i][0][2]}} ${numbers[i][1][0]}${numbers[i][1][1]}^{${numbers[i][1][2]}}`; } if(i % 2 === 0){ result += " - "; } else{ result += " + "; } i += 1; } equation.innerHTML += `
= ${result}]₀^∞`; equation.innerHTML += "
Applying the limits:
"; result = result.replace(/x/g,"∞"); equation.innerHTML += `(${result}) - `; result = result.replace(/∞/g,"(0)"); equation.innerHTML += `(${result})
`; result = result.replace(/$0$/g,"∞"); resultList = result.split(/\+|-/); resultList.pop(); i = 0; while(i < resultList.length){ if(resultList[i].length == 13){ delete resultList[i]; } else{ resultList[i] = "∞ (0)"; } i += 1; } resultList[resultList.length-1] = "0"; result = ""; i = 0; var x = 0; while(i < resultList.length){ if(i == resultList.length - 1){ result += resultList[i]; break; } else if(i % 2 === 0){ result += resultList[i]; } else if(i % 2 == 1){ i += 1; continue; } if(x % 2 === 0){ result += " - "; x += 1; } else{ x += 1; result += " + "; } i += 1; } equation.innerHTML += `= (${result}) - `; i = 1; x = z - 1; while(result.includes("0")){ result = result.replace("0",i); if(i == 1){ i = z; } else{ i *= x; x -= 1; } } result = result.replace(/∞/g,"0"); result = `(${result})`; temp = result; equation.innerHTML += `${result}
`; result = result.replace(/$/g, ""); result = result.replace(/0/g, "∞"); result = result.replace(/[0-9]+$/gi,0); result = result.replace(/$[0-9]+$/g,0); result = result.replace(/∞ 0/g,"0"); equation.innerHTML += `= (${result})`; result = result.slice(0,-1); result += fact(z); result += ")"; result = "(" + result; equation.innerHTML += ` - ${result}
`; if(z % 2 === 0){ equation.innerHTML += `= ${fact(z)}`; } else{ equation.innerHTML += `= 0 - -${fact(z)} = ${fact(z)}`; } equation.innerHTML += `
∫₀^∞ x^${z} e^-x dx = ${fact(z)}`; }; function fact(x){ if(x == 1){ return x; } else{ return x * fact(x-1); } }

Gamma Function V2