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ln(tg(pi*x/2))
Plotting a function graph/ ln(tg(pi*x/2))

Graphing y = ln(tg(pi*x/2))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /   /pi*x\\
f(x) = log|tan|----||
          \   \ 2  //
f(x)=log⁡(tan⁡(πx2))f{\left(x \right)} = \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} 
f = log(tan(pi*x/2))
The graph of the function
0-80-60-40-2020406080-5050
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
log⁡(tan⁡(πx2))=0\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=12x_{1} = \frac{1}{2}
Numerical solution
x1=82.5x_{1} = 82.5
x2=28.5x_{2} = 28.5
x3=−99.5x_{3} = -99.5
x4=−49.5x_{4} = -49.5
x5=−31.5x_{5} = -31.5
x6=84.5x_{6} = 84.5
x7=22.5x_{7} = 22.5
x8=66.5x_{8} = 66.5
x9=−85.5x_{9} = -85.5
x10=78.5x_{10} = 78.5
x11=−51.5x_{11} = -51.5
x12=−59.5x_{12} = -59.5
x13=70.5x_{13} = 70.5
x14=−37.5x_{14} = -37.5
x15=−47.5x_{15} = -47.5
x16=98.5x_{16} = 98.5
x17=−87.5x_{17} = -87.5
x18=10.5x_{18} = 10.5
x19=−55.5x_{19} = -55.5
x20=−1.5x_{20} = -1.5
x21=−89.5x_{21} = -89.5
x22=34.5x_{22} = 34.5
x23=−65.5x_{23} = -65.5
x24=62.5x_{24} = 62.5
x25=58.5x_{25} = 58.5
x26=−93.5x_{26} = -93.5
x27=−83.5x_{27} = -83.5
x28=16.5x_{28} = 16.5
x29=−39.5x_{29} = -39.5
x30=96.5x_{30} = 96.5
x31=−79.5x_{31} = -79.5
x32=−11.5x_{32} = -11.5
x33=−43.5x_{33} = -43.5
x34=4.5x_{34} = 4.5
x35=−19.5x_{35} = -19.5
x36=100.5x_{36} = 100.5
x37=12.5x_{37} = 12.5
x38=52.5x_{38} = 52.5
x39=80.5x_{39} = 80.5
x40=−97.5x_{40} = -97.5
x41=−73.5x_{41} = -73.5
x42=−67.5x_{42} = -67.5
x43=2.5x_{43} = 2.5
x44=8.5x_{44} = 8.5
x45=92.5x_{45} = 92.5
x46=−21.5x_{46} = -21.5
x47=64.5x_{47} = 64.5
x48=32.5x_{48} = 32.5
x49=56.5x_{49} = 56.5
x50=48.5x_{50} = 48.5
x51=−13.5x_{51} = -13.5
x52=−41.5x_{52} = -41.5
x53=42.5x_{53} = 42.5
x54=6.5x_{54} = 6.5
x55=90.5x_{55} = 90.5
x56=−17.5x_{56} = -17.5
x57=−7.5x_{57} = -7.5
x58=−29.5x_{58} = -29.5
x59=24.5x_{59} = 24.5
x60=44.5x_{60} = 44.5
x61=68.5x_{61} = 68.5
x62=−61.5x_{62} = -61.5
x63=38.5x_{63} = 38.5
x64=86.5x_{64} = 86.5
x65=−91.5x_{65} = -91.5
x66=50.5x_{66} = 50.5
x67=−15.5x_{67} = -15.5
x68=−81.5x_{68} = -81.5
x69=74.5x_{69} = 74.5
x70=−63.5x_{70} = -63.5
x71=−33.5x_{71} = -33.5
x72=−35.5x_{72} = -35.5
x73=−23.5x_{73} = -23.5
x74=−69.5x_{74} = -69.5
x75=40.5x_{75} = 40.5
x76=−27.5x_{76} = -27.5
x77=−9.5x_{77} = -9.5
x78=72.5x_{78} = 72.5
x79=36.5x_{79} = 36.5
x80=−25.5x_{80} = -25.5
x81=−77.5x_{81} = -77.5
x82=88.5x_{82} = 88.5
x83=−5.5x_{83} = -5.5
x84=−57.5x_{84} = -57.5
x85=60.5x_{85} = 60.5
x86=30.5x_{86} = 30.5
x87=−95.5x_{87} = -95.5
x88=26.5x_{88} = 26.5
x89=18.5x_{89} = 18.5
x90=46.5x_{90} = 46.5
x91=54.5x_{91} = 54.5
x92=−3.5x_{92} = -3.5
x93=−75.5x_{93} = -75.5
x94=94.5x_{94} = 94.5
x95=−53.5x_{95} = -53.5
x96=76.5x_{96} = 76.5
x97=20.5x_{97} = 20.5
x98=−71.5x_{98} = -71.5
x99=−45.5x_{99} = -45.5
x100=14.5x_{100} = 14.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(tan(pi*x/2)).
log⁡(tan⁡(π0⋅12))\log{\left(\tan{\left(\pi 0 \cdot \frac{1}{2} \right)} \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
π(tan⁡2(πx2)+1)2tan⁡(πx2)=0\frac{\pi \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{2 \tan{\left(\frac{\pi x}{2} \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
π2(2−tan⁡2(πx2)+1tan⁡2(πx2))(tan⁡2(πx2)+1)4=0\frac{\pi^{2} \left(2 - \frac{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}{4} = 0
Solve this equation
The roots of this equation
x1=−12x_{1} = - \frac{1}{2}
x2=12x_{2} = \frac{1}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,−12]∪[12,∞)\left(-\infty, - \frac{1}{2}\right] \cup \left[\frac{1}{2}, \infty\right)
Convex at the intervals
[−12,12]\left[- \frac{1}{2}, \frac{1}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(tan⁡(πx2))=lim⁡x→−∞log⁡(tan⁡(πx2))\lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞log⁡(tan⁡(πx2))y = \lim_{x \to -\infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}
lim⁡x→∞log⁡(tan⁡(πx2))=lim⁡x→∞log⁡(tan⁡(πx2))\lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞log⁡(tan⁡(πx2))y = \lim_{x \to \infty} \log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(tan(pi*x/2)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(tan⁡(πx2))x)=lim⁡x→−∞(log⁡(tan⁡(πx2))x)\lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(log⁡(tan⁡(πx2))x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)
lim⁡x→∞(log⁡(tan⁡(πx2))x)=lim⁡x→∞(log⁡(tan⁡(πx2))x)\lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(log⁡(tan⁡(πx2))x)y = x \lim_{x \to \infty}\left(\frac{\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
log⁡(tan⁡(πx2))=log⁡(−tan⁡(πx2))\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}
- No
log⁡(tan⁡(πx2))=−log⁡(−tan⁡(πx2))\log{\left(\tan{\left(\frac{\pi x}{2} \right)} \right)} = - \log{\left(- \tan{\left(\frac{\pi x}{2} \right)} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = ln(tg(pi*x/2))
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